Variational characterizations of $\xi$-submanifolds in the Eulicdean space ${{\mathbb {R}}}^{m+p}$

Abstract

$\xi$-submanifold in the Euclidean space ${{\mathbb {R}}}^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in ${{\mathbb {R}}}^{m+p}$. It is also a generalization of the $\lambda$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $\xi$-submanifolds are established. First, it is shown that a submanifold in ${{\mathbb {R}}}^{m+p}$ is a $\xi$-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $\big ({{\mathbb {R}}}^{m+p},e^{-\frac{1}{m}|x|^2}\left\langle \cdot ,\cdot \right\rangle \big )$; then, two generalized weighted volume functionals $V_\xi$ and ${{\bar{V}}}_\xi$ are defined and it is proved that $\xi$-submanifolds can be characterized as the critical points of these two functionals; also, the corresponding second variation formulas are computed. Finally, we introduce the VP-variations and the corresponding W-stability for $\xi$-submanifolds which are then systematically studied. As the main result, it is proved that m-planes are the only complete, W-stable and properly immersed $\xi$-submanifolds with flat normal bundle.

1 Introduction

Let $x: M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be an m-dimensional submanifold in the $(m+p)$-dimensional Euclidean space ${{\mathbb {R}}}^{m+p}$ with the second fundamental form h. Then, x is called a self-shrinker to the mean curvature flow if its mean curvature vector field $H:={\mathrm{tr\,}}h$ satisfies

$$\begin{aligned} H+x^{\bot }=0, \end{aligned}$$

(1.1)

where $x^\bot$ is the orthogonal projection of the position vector x to the normal space $T^\bot M^m$ of x.

It is well known that the self-shrinker plays an important role in the study of the mean curvature flow. In fact, self-shrinkers correspond to self-shrinking solutions to the mean curvature flow and describe all possible Type I singularities of the flow. Up to now, there have been a plenty of research papers on self-shrinkers and on the asymptotic behavior of the flow. For details of this see, for example, [1,2,3,4,5,6, 8, 12,13,14,15,16,17, 19,20,24, 28] and references therein. In particular, the following result is well known (see Corollary 3.2 in Sect. 3):

An immersion$x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$is a self-shrinker if and only if it is minimal when viewed as a submanifold of the Gaussian space$({{\mathbb {R}}}^{m+p},e^{-\frac{|x|^2}{m}}\left\langle \cdot ,\cdot \right\rangle )$.

In March, 2014, Cheng and Wei formally introduced ([9], finally revised in May, 2015) the definition of $\lambda$-hypersurface of weighted volume-preserving mean curvature flow in Euclidean space, giving a natural generalization of self-shrinkers in the hypersurface case. According to [9], a hypersurface $x: M^{m}\rightarrow {{\mathbb {R}}}^{m+1}$ is called a $\lambda$-hypersurface if its (scalar-valued) mean curvature H satisfies

$$\begin{aligned} H+\langle x, N\rangle =\lambda \end{aligned}$$

(1.2)

for some constant $\lambda$, where N is the unit normal vector of x. They also found some variational characterizations for those new kind of hypersurfaces, proving that a hypersurfacexis a$\lambda$-hypersurface if and only if it is the critical point of the weighted area functional${{\mathcal {A}}}$preserving the weighted volume functional${{\mathcal {V}}}$ where for any $x_0\in {{\mathbb {R}}}^{m+1}$ and $t_0\in {{\mathbb {R}}}$,

$$\begin{aligned} {{\mathcal {A}}}(t)=\int _M e^{-\frac{|x(t)-x_0|^2}{2t_0}}\mathrm{d}\mu ,\quad {{\mathcal {V}}}(t)=\int _M\left\langle x(t)-x_0,N\right\rangle e^{-\frac{|x(t)-x_0|^2}{2t_0}}\mathrm{d}\mu . \end{aligned}$$

Meanwhile, some rigidity or classification results for $\lambda$-hypersurfaces are obtained, for example, in [7, 10] and [18]; for the rigidity theorems for space-like $\lambda$-hypersurfaces, see [26].

We should remark that this kind of hypersurfaces was also studied in [27] (arXiv preprint: Jul. 2013; formally published in 2015) where the authors considered the stable, two-sided, smooth, properly immersed solutions to the Gaussian Isoperimetric Problem, namely they studied hypersurfaces $\Sigma \subset {{\mathbb {R}}}^{m+1}$ that are second order stable critical points of minimizing the weighted area functional ${{\mathcal {A}}}_\mu (\Sigma )=\int _\Sigma e^{-|x|^2/4}d{{\mathcal {A}}_\mu }$ for compact (uniformly) normal variations that, in a sense, “preserve the weighted volume${{\mathcal {V}}}_\mu (\Sigma )=\int _\Sigma e^{-|x|^2/4}d{{\mathcal {V}}_\mu }$”. It turned out that the $\lambda$-hypersurface equation (1.2) is exactly the Euler-Lagrange equation of the variation problem in [27] of which a main result can be restated as

Hyperplanes are the only two-sided, complete and properly immersed$\lambda$-hypersurfaces in the Euclidean space that are stable under the compact normal variations “preserving the weighted volume”.

In 2015, the first author and his co-author made in [25] a natural generalization of both self-shrinkers and $\lambda$-hypersurfaces by introducing the concept of $\xi$-submanifolds and, as the main result, a rigidity theorem for Lagrangian $\xi$-submanifolds in ${{\mathbb {C}}}^2$ is proved, which is motivated by a result of [23] for Lagrangian self-shrinkers in ${{\mathbb {C}}}^2$. By definition, an immersed submanifold $x: M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is called a $\xi$-submanifold if there is a parallel normal vector field $\xi$ such that the mean curvature vector field H satisfies

$$\begin{aligned} H+x^{\bot }=\xi . \end{aligned}$$

(1.3)

We believe that if self-shrinkers and $\lambda$-hypersurfaces are taken to be parallel to minimal submanifolds and constant mean curvature hypersurfaces, respectively, then $\xi$-submanifolds are expected to be parallel to submanifolds of parallel mean curvature vector. So there should be many properties of $\xi$-submanifolds that are parallel to those of submanifolds with parallel mean curvature vectors.

In this paper, we aim at giving more characterizations of the $\xi$-submanifolds, including ones by variation method, the latter being more important since a differential equation usually needs a variational method to solve. For example, self-shrinker equation (1.1) has been exploited a lot by making use of variation formulas. As the main part of this paper, we shall systematically study the relevant stability problems for $\xi$-submanifolds, paying a particular attention on the VP-variations and the relevant W-stability.

Now, beside the various characterizations of the $\xi$-submanifolds and some instability results, the main theorem of this paper can be stated as

Theorem 1.1

(Theorem 7.3). Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a complete and properly immersed $\xi$-submanifold with flat normal bundle. Then, x is W-stable if and only if $x(M^m)$ is an m-plane.

Clearly, Theorem 1.1 generalizes the main theorem for hypersurfaces in [27] which has been stated earlier.

The following uniqueness conclusion for self-shrinkers is direct from Theorem 1.1:

Corollary 1.2

Any complete, W-stable and properly immersed self-shrinker in ${{\mathbb {R}}}^{m+p}$ with flat normal bundle must be an m-plane passing the origin.

The organization of the present paper is as follows:

In Sect. 2, we present the necessary preliminary material, including some typical examples;

In Sect. 3, we prove a theorem (Theorem 3.1) which generalizes (to $\xi$-submanifolds) a well-known result that self-shrinkers are equivalent to minimal submanifolds in the Gaussian space;

In Sect. 4, we introduce, for a given manifold $M^m$ of dimension m, two families of weighted volume functionals $V_\xi$ and ${{\bar{V}}}_\xi$ in (4.1) parametrized by ${{\mathbb {R}}}^{m+p}$-valued functions $\xi :M^m\rightarrow {{\mathbb {R}}}^{m+p}$. Then we compute the first variation formulas (Theorem 4.1) which give that $\xi$-submanifolds are exactly the critical points of $V_\xi$ and ${{\bar{V}}}_\xi$ with $\xi$ suitably chosen (Corollary 4.2). We also compute the second variation formula of both functionals for $\xi$-submanifolds (Theorem 4.3), in such a situation $V_\xi$ and ${{\bar{V}}}_\xi$ being essential the same.

In Sects. 5 and 6, we study the stability problem of $\xi$-submanifolds. After checking that, with respect to the functional $V_\xi$ or ${{\bar{V}}}_\xi$, many $\xi$-submanifolds including all the typical examples are not stable in the usual sense (Sect. 5), we define in Sect. 6 a special kind of variation for submanifolds of higher codimension, called “VP-variation,” which is a natural generalization of “volume-preserving variation” for hypersurfaces. Accordingly, we introduce “the$W_\xi$-stability” with respect to $V_\xi$ or ${{\bar{V}}}_\xi$ for higher codimensional submanifolds and then show that, among the typical examples given in Sect. 2, only the m-planes are $W_\xi$-stable (Theorem 6.1 and Theorem 6.2). In particular, we give an index estimate for the standard sphere (Theorem 6.2).

Finally, in the last section (Sect. 7), we consider the VP-variation of the standard weighted volume functional $V_w\equiv V_0$ which corresponds to a special case, i.e., $\xi =0$, of the functional $V_\xi$ or ${{\bar{V}}}_\xi$ defined in Sect. 4, and study the W-stability (i.e., $W_0$-stability, see Definition 7.1) for $\xi$-submanifolds. As the result, we first characterize $\xi$-submanifolds as critical points of $V_w$ under VP-variations (Corollary 7.2, corresponding to the conventional extremal points with conditions) and then prove our main Theorem (Theorem 1.1).

Remark 1.1

Our discussion of variation problem for $\xi$-submanifolds naturally gives a new motivation of variational characterization of the submanifolds with parallel mean curvature vectors in the Euclidean space ${{\mathbb {R}}}^{m+p}$ (see Remark 4.3 at the end of Sect. 4).

Remark 1.2

Related to the present paper, it seems natural and interesting to characterize $\xi$-submanifolds in terms of their Gauss map, just like in the study of submanifolds in ${{\mathbb {R}}}^{m+p}$ with parallel mean curvature vectors. We shall deal this kind of problems later in the sequel.

2 $\xi$-submanifolds–definition and typical examples

Let ${{\mathbb {R}}}^{m+p}$ be the $(m+p)$-dimensional Euclidean space with the standard metric $\left\langle \cdot ,\cdot \right\rangle$ and the standard connection D. Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be an immersion with the induced metric g, the second fundamental form h and the mean curvature vector $H:={\mathrm{tr\,}}_g h$. Denote by TM the tangent space of M with the Levi-Civita connection $\nabla$, and define $T^\bot M:=(x_*(TM))^\bot$ to be the normal space of x in ${{\mathbb {R}}}^{m+p}$ with the normal connection $D^\bot$.

Definition 2.1

($\xi$-submanifolds, [25]). The immersed submanifold $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is called a $\xi$-submanifold if the normal vector field

$$\begin{aligned} \xi :=H+x^\bot \end{aligned}$$

(2.1)

is parallel in $T^\bot M$, namely $D^\bot \xi \equiv 0$.

So, self-shrinkers of the mean curvature flow are a special kind of $\xi$-submanifolds with $\xi =0$.

The following are some typical examples of $\xi$-submanifolds:

Example 2.1

(The $\xi$-curves).

Let $x:(a,b)\rightarrow {{\mathbb {R}}}^{1+p}$ be a unit-speed smooth curve (that is, with an arc-length parameter s). Denote by $\{T,e_\alpha :\ 2\le \alpha \le 1+p\}$ the Frenet frame with $T:=\dot{x}\equiv \frac{\partial x}{\partial s}$ being the unit tangent vector, and $\kappa _i$ the i-th curvature, $i=1,\ldots , p$. Then, we have the following Frenet formula:

$$\begin{aligned} \dot{T}=\kappa _1e_2,\ \dot{e}_2=-\kappa _1T+\kappa _2e_3,\ \cdots ,\ \dot{e}_p=-\kappa _{p-1}e_{p-1}+\kappa _pe_{p+1},\ \dot{e}_{1+p}=-\kappa _pe_p. \end{aligned}$$

(2.2)

In particular, if there exists some i such that $\kappa _i\equiv 0$, then it must hold that $\kappa _j\equiv 0$ for all $j>i$. Sometimes we call $\kappa :=\kappa _1$ and $\tau :=\kappa _2$ the curvature and the (first) torsion of x. Now the definition Eq. (2.1) becomes $\left( \frac{\mathrm{d} }{\mathrm{d} s}(\dot{T}+x-\left\langle x,T\right\rangle T)\right) ^\bot \equiv 0$ which, by (2.2), is equivalent to

$$\begin{aligned} {{\dot{\kappa }}}_1-\kappa _1\left\langle x,T\right\rangle \equiv 0,\quad \kappa _1\kappa _2\equiv 0. \end{aligned}$$

(2.3)

It follows that

xis a$\xi$-curve if and only if it is a plane curve with the curvature$\kappa$satisfying

$$\begin{aligned} {{\dot{\kappa }}}-\kappa \left\langle x,\dot{x}\right\rangle \equiv 0. \end{aligned}$$

(2.4)

In particular,

xis a self-shrinker if and only if it is a plane curve with the curvature$\kappa$satisfying

$$\begin{aligned} \kappa _r+\left\langle x,N\right\rangle \equiv 0, \end{aligned}$$

(2.5)

where $\kappa _r$ is the relative curvature and $N:=\pm e_2$ is the unit normal of x pointing the left of T. Note that curves in the plane satisfying (2.5) are classified by U. Abresch and J. Langer in [1] which are now known as Abresch–Langer curves (see [23]).

Example 2.2

(The m-planes not necessarily passing through the origin).

An m-plane $x:P^m\rightarrow {{\mathbb {R}}}^{m+p}$ ($p\ge 0$) is by definition the inclusion map of a m-dimensional connected, complete and totally geodesic submanifold of ${{\mathbb {R}}}^{m+p}$. In other words, those $P^m$s are subplanes of dimension m in ${{\mathbb {R}}}^{m+p}$ that are not necessarily passing through the origin. Let $p_0$ be the orthogonal projection of the origin 0 onto $P^m$ and $\xi$ be the position vector of $p_0$ which is constant and is thus parallel along $P^m$. Clearly $P^m$ is a $\xi$-submanifold because $H\equiv 0$ and the tangential part $x^\top$ of x is precisely $x-\xi$.

Example 2.3

(The standard spheres centered at the origin).

For a given point $x_0\in {{\mathbb {R}}}^{m+1}$ and a positive number r. Define

$$\begin{aligned} S^m(r,x_0)=\{x\in {{\mathbb {R}}}^{m+1};\ |x-x_0|=r\}, \end{aligned}$$

the standard m-sphere in ${{\mathbb {R}}}^{m+1}$ with radius r and center $x_0$. In particular, we denote $S^m(r):=S^m(r,0)$. It is easily found that $S^m(r,x_0)$ is a $\xi$-submanifold if and only if $x_0=0$.

In fact, since $x-x_0$ is a normal vector field of length r, the normal part $x^\bot$ of x is

$$\begin{aligned} x^\bot =\frac{1}{r^2}\left\langle x,x-x_0\right\rangle (x-x_0). \end{aligned}$$

Note that $H=-\frac{m}{r^2}(x-x_0)$ is parallel. It follows that $H+x^\bot$ is parallel if and only if $x^\bot$ is. This is clearly equivalent to that $\left\langle x,dx\right\rangle \equiv 0$ which is true if and only if $x_0=0$.

Example 2.4

(Submanifolds in a sphere with parallel mean curvature vector).

Let $x:M^m\rightarrow S^{m+p}(a)\subset {{\mathbb {R}}}^{m+p+1}$ be a submanifold in the standard sphere $S^{m+p}(a)$ of radius a, which is of parallel mean curvature vector H. Then, as a submanifold of ${{\mathbb {R}}}^{m+p+1}$, x is a $\xi$-submanifold.

In fact, as the submanifold of ${{\mathbb {R}}}^{m+p+1}$, the mean curvature vector of x is ${{\bar{H}}}=\triangle x=H-\frac{m}{a^2}x$. Thus, $\xi :={{\bar{H}}}+x^\bot =H+(1-\frac{m}{a^2})x$ which is clearly parallel. In particular, $x(M^m)\subset {{\mathbb {R}}}^{m+p+1}$ is a self-shrinker if and only if $x(M^m)\subset S^{m+p}(a)$ is a minimal submanifold.

Example 2.5

(The product of $\xi$-submanifolds).

Let $x_a:M^{m_a}\rightarrow {{\mathbb {R}}}^{m_a+p_a}$, $a=1,2$, be two immersed submanifolds. Denote $m=m_1+m_2$, $p=p_1+p_2$ and $M^m=M^{m_1}\times M^{m_2}$. Then, it is not hard to show that $x:=x_1\times x_2:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is a $\xi$-submanifold if and only if both $x_1$ and $x_2$ are $\xi$-submanifolds.

In particular, for any given positive numbers $r_1,\ldots ,r_k$ ($k\ge 0$), positive integers $m_1,\ldots , m_k,n_1,\ldots ,n_l$ ($l\ge 0$, $k+l>0$) and $n\ge n_1+\cdots +n_l$, the embedding

$$\begin{aligned} x:S^{m_1}(r_1)\times \cdots \times S^{m_k}(r_k)\times P^{n_1}\times \cdots \times P^{n_l}\rightarrow {{\mathbb {R}}}^{m_1+\cdots +m_k+k+n} \end{aligned}$$

(2.6)

are all $\xi$-submanifolds.

Remark 2.1

Apart from these typical examples of $\xi$-submanifolds given above, there should certainly be other nonstandard examples. In particular, we have the so-called $\lambda$-torus constructed by Q.-M. Cheng and G. X. Wei in [11], which is among a general class of rotational $\lambda$-hypersurfaces. Precisely, we have

Theorem 2.1

([11]). For any $m\ge 2$ and $\lambda >0$, there exists an embedded rotational $\lambda$-hypersurface $x: M^m\rightarrow {{\mathbb {R}}}^{m+1}$, which has the topology of the torus ${{\mathbb {S}}}^1\times {{\mathbb {S}}}^{m-1}$.

It would be interesting if one can construct similar $\xi$-submanifolds with certain symmetry.

3 As submanifolds of the Gaussian space

As mentioned in the introduction, m-dimensional self-shrinkers of the mean curvature flow in the Euclidean space ${{\mathbb {R}}}^{m+p}\equiv ({{\mathbb {R}}}^{m+p},\left\langle \cdot ,\cdot \right\rangle )$ is equivalent to being minimal submanifolds when viewed as submanifolds in the Gaussian metric space $({{\mathbb {R}}}^{m+p}, {{\bar{g}}})$ where ${{\bar{g}}}:= e^{-\frac{|x|^2}{m}}\left\langle \cdot ,\cdot \right\rangle$. In this section, we generalize this to $\xi$-submanifolds to obtain our first characterization. In fact, we will prove a theorem which says that $\xi$-submanifolds are essentially equivalent to being submanifolds of parallel mean curvature in $({{\mathbb {R}}}^{m+p}, {{\bar{g}}})$.

For an immersion $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$, we use $(\overline{\cdots })$ to denote geometric quantities when x is taken as an immersion into $({{\mathbb {R}}}^{m+p}, {{\bar{g}}})$ that correspond those quantities $(\cdots )$ when x is taken as an immersion into $({{\mathbb {R}}}^{m+p}, \left\langle \cdot ,\cdot \right\rangle )$. So, for example, we have the induced metric ${{\bar{g}}}$, the second fundamental form ${{\bar{h}}}$ and the mean curvature ${{\bar{H}}}$, etc. To make things more clear, we would like to introduce a “modified mean curvature” for the immersion x, which is defined as $\tilde{H}=e^{-\frac{|x|^2}{2m}}{{\bar{H}}}$. Then, we have

Theorem 3.1

(The first characterization). An immersion $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is a $\xi$-submanifold if and only if its modified mean curvature $\tilde{H}$ is parallel.

Proof

Denote by ${{\bar{D}}}$ the Levi-Civita connections of $({{\mathbb {R}}}^{m+p}, {{\bar{g}}})$. For any given frame field $\{e_A;\ A=1,2\cdots ,m+p\}$, the corresponding connection coefficients of the standard connection D and ${{\bar{D}}}$ are, respectively, denoted by $\Gamma ^C_{AB}$ and ${{\bar{\Gamma }}}^C_{AB}$ with $A,B,C,\ldots =1,2,\ldots m+p$. Then by the Koszul formula, we find

$$\begin{aligned} {{\bar{\Gamma }}}^C_{AB}=\Gamma ^C_{AB}+\frac{1}{m}\left( g(x,e_D)g_{AB}g^{CD}-g(x,e_A)\delta ^C_B-g(x,e_B)\delta ^C_A\right) , \end{aligned}$$

(3.1)

or equivalently,

$$\begin{aligned} {{\bar{D}}}_{e_B}e_A=D_{e_B}e_A+\frac{1}{m}(g_{AB}x-g(x,e_A)e_B-g(x,e_B)e_A). \end{aligned}$$

(3.2)

Now given an immersion $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$, the induced metric on $M^m$ by x of the ambient metric ${{\bar{g}}}$ will still be denoted by ${{\bar{g}}}$. Choose a frame field $\{e_i,e_\alpha \}$ along x such that $e_i$, $i=1,2,\ldots ,m$, are tangent to $M^m$ and $e_\alpha$, $\alpha =m+1,\ldots ,m+p$ are normal to $x_*(TM^m)$ satisfying $\left\langle e_\alpha ,e_\beta \right\rangle \equiv g(e_\alpha ,e_\beta )=\delta _{\alpha \beta }$. Then by the Gauss formula and (3.1) or (3.2), we find the relation between the second fundamental forms ${{\bar{h}}}$ and h is as follows:

$$\begin{aligned} {{\bar{h}}}_{ij}\equiv {{\bar{h}}}(e_i,e_j)=\left( {{\bar{D}}}_{e_j}e_i\right) ^\bot =h_{ij}+\frac{1}{m} x^\bot g_{ij} \end{aligned}$$

(3.3)

where $h_{ij}=h(e_i,e_j)=\left( D_{e_j}e_i\right) ^\bot$. It follows that the mean curvature vectors satisfy

$$\begin{aligned} {{\bar{H}}}\equiv {{\bar{g}}}^{ij}{{\bar{h}}}_{ij}=e^{\frac{|x|^2}{m}}(H+x^\bot ). \end{aligned}$$

(3.4)

Now we compute the covariant derivative of the modified mean curvature $\tilde{H}\equiv e^{-\frac{|x|^2}{2m}}{{\bar{H}}}$ with respect to the normal connection ${{\bar{D}}}^\bot$. First we note that, since ${{\bar{g}}}$ is conformal to $\left\langle \cdot ,\cdot \right\rangle$ on ${{\mathbb {R}}}^{m+p}$, $\{e_\alpha \}$ which satisfies $\left\langle e_\alpha ,e_\beta \right\rangle =\delta _{\alpha \beta }$ remains a normal frame field of x considered as the immersion into $({{\mathbb {R}}}^{m+p},{{\bar{g}}})$, not orthonormal anymore. Thus, we can write

$$\begin{aligned} \tilde{H}=\sum \tilde{H}^\alpha e_{\alpha }\ \text { with } \ \tilde{H}^\alpha =e^{\frac{|x|^2}{2m}}(H^\alpha +\left\langle x,e_\alpha \right\rangle ) \end{aligned}$$

where $H=\sum H^\alpha e_\alpha$. Note that by (3.1),

$$\begin{aligned} {{\bar{\Gamma }}}^\alpha _{\beta i}=\Gamma ^\alpha _{\beta i}-\frac{1}{m}\left\langle x,e_i\right\rangle \delta ^\alpha _\beta , \quad \forall \alpha ,\beta , i. \end{aligned}$$

It follows that, for each $\alpha =m+1,\ldots ,m+p$,

$$\begin{aligned} \left( {{\bar{D}}}^\bot _{e_i}\tilde{H}\right) ^\alpha&=e_i(\tilde{H}^\alpha )+\tilde{H}^\beta {{\bar{\Gamma }}}^\alpha _{\beta i}\\&=e_i\big (e^{\frac{|x|^2}{2m}}\big )(H^\alpha +\left\langle x,e_\alpha \right\rangle )+e^{\frac{|x|^2}{2m}}(e_i(H^\alpha )+e_i\left\langle x,e_\alpha \right\rangle )\\&\quad +e^{\frac{|x|^2}{2m}}(H^\beta +\left\langle x,e_\beta \right\rangle )\left( \Gamma ^\alpha _{\beta i}-\frac{1}{m}\left\langle x,e_i\right\rangle \delta ^\alpha _\beta \right) \\&=e^{\frac{|x|^2}{2m}}(e_i(H^\alpha )+e_i\left\langle x,e_\alpha \right\rangle +H^\beta \Gamma ^\alpha _{\beta i}+\left\langle x,e_\beta \right\rangle \Gamma ^\alpha _{\beta i})\\&=e^{\frac{|x|^2}{2m}}\big (D^\bot _{e_i}(H+x^\bot )\big )^\alpha , \end{aligned}$$

where ${{\bar{D}}}^\bot$, $D^\bot$ denote the induced normal connections accordingly. Thus, Theorem 3.1 is proved. $\square$

The following conclusion is direct by (3.4):

Corollary 3.2

An immersion $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is a self-shrinker if and only if it is minimal when viewed as a submanifold of the Gaussian space $({{\mathbb {R}}}^{m+p},{{\bar{g}}})$.

4 Variational characterizations

In this section, we first define two functionals and derive the corresponding first and second variation formulas, aiming to establish variational characterizations of the $\xi$-submanifolds.

For a given manifold $M\equiv M^m$ of dimension m, define

$$\begin{aligned} {{\mathcal {M}}}:=\{\text {all the immersions }x:M^m\rightarrow {{\mathbb {R}}}^{m+p}\} \end{aligned}$$

and let $\xi :M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a vector-valued function on the manifold $M^m$. Then, we can naturally introduce, as follows, two kinds of interesting functionals $V_\xi$ and ${{\bar{V}}}_\xi$ on ${\mathcal {M}}$ which are parametrized by $\xi$:

$$\begin{aligned} V_\xi (x):=\int _M e^{-f_x}\mathrm{d}V_x,\quad {{\bar{V}}}_\xi (x)=\int _M e^{-{{\bar{f}}}_x}\mathrm{d}V_x,\quad x\in {{\mathcal {M}}}, \end{aligned}$$

(4.1)

where for any $p\in M^m$, $f_x(p):=\frac{1}{2}|x(p)-\xi (p)|^2$, ${{\bar{f}}}_x(p)=f_x(p)-\frac{1}{2}|\xi (p)|^2$ and $\mathrm{d}V_x$ is the volume element of the induced metric $g_x$ of x.

Remark 4.1

(1) These two functionals $V_\xi$ and ${{\bar{V}}}_\xi$ are both of weighted volumes in a sense since, for example, the weighted volume element $e^{-\frac{1}{2}|x-\xi |^2} \mathrm{d}V_x$ corresponding to the first one can be viewed as induced from an unnormalized “general Gaussian measure” on the ambient Euclidean space ${{\mathbb {R}}}^{m+p}$ with “mean” $\xi$. Note that when $\xi$ is constant as in the case of m-planes, $\left( \frac{1}{\sqrt{2\pi }}\right) ^{m+p}e^{-f_x}\mathrm{d}V_{{{\mathbb {R}}}^{m+p}}$ is nothing but the usual generalized Gaussian measure with the mean $\xi$ (and the variance$\sigma ^2\equiv 1$)¹; meanwhile, the functional ${{\bar{V}}}_\xi$ is clearly a new weighted volume obtained from $V_\xi$ by just adding a new weight $e^{\frac{1}{2}|\xi |^2}$. Also, the weight function $e^{-f_x}$ or $e^{-{{\bar{f}}}_x}$ naturally has a close relation with the definition of the Hermitian Polynomials (see, for example, [14] and [15]). These polynomials will also be used later in our stability discussion in Sect. 5.

(2) All of the typical $\xi$-submanifolds (that is, m-planes $P^m$, standard m-spheres $S^m(r)$) and their products (2.6) have finite values for both the functionals $V_\xi$ and ${{\bar{V}}}_\xi$, where $\xi$ is chosen to be $H+x^\bot$.

Now let $x\in {{\mathcal {M}}}$ be fixed with the induced Riemannian metric $g:=x^*\left\langle \cdot ,\cdot \right\rangle$ and suppose that $F:M\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ is a variation of x with $\eta :=F_*(\frac{\partial }{\partial t})|_{t=0}$ being the corresponding variation vector field. For $p\in M$, $t\in (-\varepsilon ,\varepsilon )$, denote

$$\begin{aligned} x_t(p)=F(p,t),\quad \frac{\partial F}{\partial t}=F_*\left( \frac{\partial }{\partial t}\right) ,\quad \frac{\partial F}{\partial u^i}=F_*(\frac{\partial }{\partial u^i})\equiv (x_t)_*\left( \frac{\partial }{\partial u^i}\right) \end{aligned}$$

where $(u^i)$ is a local coordinates on M. We always assume that, for each $t\in (-\varepsilon ,\varepsilon )$, $x_t:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is an immersion, that is, $x_t\in {{\mathcal {M}}}$, $t\in (-\varepsilon ,\varepsilon )$.

Definition 4.1

(Compact variation). A variation $F:M\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ is called compactly supported, or simply compact, if there exists a relatively compact open domain B such that, for each $t\in (-\varepsilon ,\varepsilon )$, the support set $\overline{\{p\in M^m;\ \frac{\partial F}{\partial t}(p)\ne 0\}}$ of the vector field $\frac{\partial F}{\partial t}$ is contained in B.

Denote $f_t=f_{x_t}$, ${{\bar{f}}}_t={{\bar{f}}}_{x_t}$ and

$$\begin{aligned} \Gamma _0(T^\bot (M))=\{\text {all smooth normal vector fields } \eta \text { of } x \text { with compact support}\}. \end{aligned}$$

Theorem 4.1

(The first variation formula). Let F be a compact variation of x. Then,

$$\begin{aligned} V_\xi '(t)&=-\int _M \left\langle (H_{t}+x_{t}^\bot -\xi )+\nabla ^t\left( \left\langle x_t,\xi \right\rangle -\frac{1}{2}|\xi |^2\right) ,\frac{\partial F}{\partial t}\right\rangle e^{-f_t}\mathrm{d}V_t, \end{aligned}$$

(4.2)

$$\begin{aligned} {{\bar{V}}}_\xi '(t)&=-\int _M \left\langle (H_{t}+x_{t}^\bot -\xi )+\nabla ^t\left\langle x_t,\xi \right\rangle ,\frac{\partial F}{\partial t}\right\rangle e^{-{{\bar{f}}}_t}\mathrm{d}V_t, \end{aligned}$$

(4.3)

where $H_t$ is the mean curvature vector of the immersion $x_t$, $\nabla ^t$ is the gradient operator of the induced metric $g_{x_t}$ and $\mathrm{d}V_t=\mathrm{d}V_{x_t}$.

In particular, if F is a normal variation of x, that is, $\eta \in \Gamma _0(T^\bot (M))$, then

$$\begin{aligned} V_\xi '(0)&=-\int _M\left\langle (H+x^\bot -\xi ),\eta \right\rangle e^{-f_0}\mathrm{d}V, \end{aligned}$$

(4.4)

$$\begin{aligned} {{\bar{V}}}_\xi '(0)&=-\int _M\left\langle (H+x^\bot -\xi ),\eta \right\rangle e^{-{{\bar{f}}}_0}\mathrm{d}V. \end{aligned}$$

(4.5)

Proof

From now on, we shall always write f for $f_x$ or $f_t$ in the computation. It is well known that

$$\begin{aligned} \frac{\partial }{\partial t}\mathrm{d}V_t&=\left( {\mathrm{div}}\left( \frac{\partial F}{\partial t}\right) ^\top -\left\langle H_t,\frac{\partial F}{\partial t}\right\rangle \right) \mathrm{d}V_t\\&=\left( \left( g_t^{ij}\left\langle \frac{\partial F}{\partial u^i},\frac{\partial F}{\partial t}\right\rangle \right) _{,j}-\left\langle H_t,\frac{\partial F}{\partial t}\right\rangle \right) \mathrm{d}V_t. \end{aligned}$$

Furthermore

$$\begin{aligned} \frac{\partial }{\partial t}e^{-f}=-e^{-f}\frac{\partial f}{\partial t}=-e^{-f}\left\langle x_t-\xi ,\frac{\partial F}{\partial t}\right\rangle . \end{aligned}$$

Thus by using the divergence theorem, we find

$$\begin{aligned} V_\xi '(t)&\left. =\int _M\frac{\partial }{\partial t}\left( e^{-f}\mathrm{d}V_t\right) =\int _M\left( \frac{\partial }{\partial t}e^{-f}\right) \mathrm{d}V_t+e^{-f}\frac{\partial }{\partial t}\mathrm{d}V_t\right) \\&=\int _M\left( -e^{-f}\left\langle x_t-\xi ,\frac{\partial F}{\partial t}\right\rangle +e^{-f}\left( \left( g_t^{ij}\left\langle \frac{\partial F}{\partial u^i},\frac{\partial F}{\partial t}\right\rangle \right) _{,j}-\left\langle H_t,\frac{\partial F}{\partial t}\right\rangle \right) \right) \mathrm{d}V_t\\&\left. =-\int _M\left( \left\langle H_t+ x_t^\bot -\xi ,\frac{\partial F}{\partial t}\right\rangle +g_t^{ij}\frac{\partial }{\partial u^j}\left( \left\langle x_t,\xi \right\rangle -\frac{1}{2}|\xi |^2\right) \frac{\partial F}{\partial u^i},\frac{\partial F}{\partial t}\right\rangle \right) e^{-f}\mathrm{d}V_t\\&=-\int _M\left( \left\langle (H_t+ x_t^\bot -\xi )+\nabla ^t\left( \left\langle x_t,\xi \right\rangle -\frac{1}{2}|\xi |^2\right) ,\frac{\partial F}{\partial t}\right\rangle \right) e^{-f}\mathrm{d}V_t, \end{aligned}$$

which gives (4.2). The other formula (4.3) is derived in the same way. $\square$

Corollary 4.2

(Variational characterizations). An immersion $x\in {{\mathcal {M}}}$ is a $\xi$-submanifold if and only if there exists a parallel normal vector field $\xi \in \Gamma (T^\bot M)$ such that x is the critical point of both the functionals $V_\xi$, ${{\bar{V}}}_\xi$ for all the compact normal variations of x.

To find the second variational formulas, we suppose that x is a $\xi$-submanifold, that is, $H+x^\bot =\xi$, where $\xi$ is a parallel normal vector of x. In particular, $|\xi |^2$ is a constant. Note that in this case, the two functionals $V_\xi$ and ${{\bar{V}}}_\xi$ are essentially the same. So in what follows we only need to consider $V_\xi$.

Suppose that F is a compact normal variation of x. Then from (4.2), we have

$$\begin{aligned} V_\xi ''(0)&=-\int _M \left\langle D_{\frac{\partial }{\partial t}}\left( (H_{t}+x_{t}^\bot -\xi )+\nabla ^t\left\langle x_t,\xi \right\rangle \right) ,\frac{\partial F}{\partial t}\right\rangle |_{t=0}e^{-f}\mathrm{d}V\nonumber \\&\quad -\int _M \left\langle \nabla ^t\left\langle x_t,\xi \right\rangle , D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial t}\right\rangle |_{t=0} e^{-f}\mathrm{d}V\nonumber \\&=-\int _M \left\langle D_{\frac{\partial }{\partial t}}\left( (H_{t}+x_{t}^\bot -\xi )+\nabla ^t\left\langle x_t,\xi \right\rangle \right) |_{t=0},\eta \right\rangle e^{-f}\mathrm{d}V\nonumber \\&\quad -\int _M \left\langle \nabla \left\langle x,\xi \right\rangle , D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial t}|_{t=0}\right\rangle e^{-f}\mathrm{d}V. \end{aligned}$$

(4.6)

Since

$$\begin{aligned} H_{t}=(g_{t})^{ij}h_t\left( \frac{\partial }{\partial u^i},\frac{\partial }{\partial u^j}\right) =(g_{t})^{ij}\left( D_{\frac{\partial }{\partial u^j}}\frac{\partial F}{\partial u^i}-(x_{t})_{*}\nabla _{\frac{\partial }{\partial u^j}}^{t}\frac{\partial }{\partial u^i}\right) , \end{aligned}$$

we have

$$\begin{aligned} D_{\frac{\partial }{\partial t}}H_{t}=\frac{\partial }{\partial t}(g_{t})^{ij}h_t\left( \frac{\partial }{\partial u^i},\frac{\partial }{\partial u^j}\right) +(g_{t})^{ij} D_\frac{\partial }{\partial t}\left( D_{\frac{\partial }{\partial u^j}}\frac{\partial F}{\partial u^i}-(x_{t})_{*}\nabla _{\frac{\partial }{\partial u^j}}^{t}\frac{\partial }{\partial u^i}\right) . \end{aligned}$$

(4.7)

On the other hand,

$$\begin{aligned} \left( \frac{\partial }{\partial t}(g_{t})^{ij}\right) |_{t=0}&=-\left( \left( (g_{t})^{ik}(g_{t})^{jl}\left\langle D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial u^k},\frac{\partial F}{\partial u^l}\right\rangle + \left\langle \frac{\partial F}{\partial u^k}, D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial u^l}\right\rangle \right) \right) |_{t=0}\\&=-g^{ik}g^{jl}\left( \frac{\partial }{\partial u^k}\left\langle \frac{\partial F}{\partial t},\frac{\partial F}{\partial u^l}\right\rangle -\left\langle \frac{\partial F}{\partial t} , D_{\frac{\partial }{\partial u^k}}\frac{\partial F}{\partial u^l}\right\rangle \right) |_{t=0}\\&\quad -g^{ik}g^{jl}\left( \frac{\partial }{\partial u^l}\left\langle \frac{\partial F}{\partial t},\frac{\partial F}{\partial u^k}\right\rangle -\left\langle \frac{\partial F}{\partial t} , D_{\frac{\partial }{\partial u^l}}\frac{\partial F}{\partial u^k}\right\rangle \right) |_{t=0}\\&=g^{ik}g^{jl}\left\langle h(\frac{\partial }{\partial u^k},\frac{\partial }{\partial u^l}),\eta \right\rangle +g^{ik}g^{jl}\left\langle h(\frac{\partial }{\partial u^l},\frac{\partial }{\partial u^k}),\eta \right\rangle , \end{aligned}$$

and by the flatness of ${{\mathbb {R}}}^{m+p}$,

$$\begin{aligned} D_\frac{\partial }{\partial t} D_{\frac{\partial }{\partial u^j}}\frac{\partial F}{\partial u^i}|_{t=0}&=D_\frac{\partial }{\partial u^j} D_\frac{\partial }{\partial t}\frac{\partial F}{\partial u^i} + D_{\left[ \frac{\partial }{\partial t},\frac{\partial }{\partial u^j}\right] }\frac{\partial F}{\partial u^i}|_{t=0}\\&=D_{\frac{\partial }{\partial u^j}}\left( D_\frac{\partial }{\partial u^i}^\bot \eta -x_*\left( A_\eta \frac{\partial }{\partial u^i}\right) \right) \\&=D_{\frac{\partial }{\partial u^j}}^\bot D_\frac{\partial }{\partial u^i}^\bot \eta -h\left( \frac{\partial }{\partial u^j},A_\eta \left( \frac{\partial }{\partial u^i}\right) \right) \\&\quad -x_{*}\left( A_{D_{\frac{\partial }{\partial u^i}}^\bot \eta }\frac{\partial }{\partial u^j}\right) -x_{*}\left( \nabla _{\frac{\partial }{\partial u^j}}\left( A_\eta \frac{\partial }{\partial u^i}\right) \right) \end{aligned}$$

where $A_\eta$ is the Weingarten operator of x with respect to the variation vector $\eta$. Moreover,

$$\begin{aligned} D_\frac{\partial }{\partial t}\left( \left( x_{t}\right) _{*}\nabla _{\frac{\partial }{\partial u^j}}^{t}\frac{\partial }{\partial u^i}\right) |_{t=0}&= D_\frac{\partial }{\partial t}\left( \left( \Gamma _{t}\right) _{ij}^{k}\left( x_{t}\right) _{*}\frac{\partial }{\partial u^k}\right) |_{t=0}\\&=\frac{\partial }{\partial t}((\Gamma _{t})_{ij}^{k})|_{t=0}x_*\frac{\partial }{\partial u^k}+\Gamma _{ij}^{k} D_\frac{\partial }{\partial t}\left( \frac{\partial F}{\partial u^k}\right) |_{t=0}\\&=\frac{\partial }{\partial t}((\Gamma _{t})_{ij}^{k})|_{t=0}x_*\frac{\partial }{\partial u^k}+D_{\nabla _{\frac{\partial }{\partial u^j}}\frac{\partial }{\partial u^i}}\eta . \end{aligned}$$

It then follows that

$$\begin{aligned}&\left\langle \frac{\partial }{\partial t}\left( g_{t}\right) ^{ij} h_t\left( \frac{\partial }{\partial u^i},\frac{\partial }{\partial u^j}\right) |_{t=0},\eta \right\rangle =2g^{ik}g^{jl}\left\langle h\left( {\frac{\partial }{\partial u^k},{\frac{\partial }{\partial u^l}}}\right) ,\eta \right\rangle \left\langle h\left( {\frac{\partial }{\partial u^i},{\frac{\partial }{\partial u^j}}}\right) ,\eta \right\rangle , \end{aligned}$$

(4.8)

$$\begin{aligned}&g^{ij}\left\langle D_\frac{\partial }{\partial t} D_{\frac{\partial }{\partial u^j}}\frac{\partial F}{\partial u^i}|_{t=0},\eta \right\rangle =g^{ij}\left( \left\langle D_{\frac{\partial }{\partial u^j}}^\bot D_\frac{\partial }{\partial u^i}^\bot \eta -h\left( \frac{\partial }{\partial u^j},A_\eta \left( \frac{\partial }{\partial u^i}\right) \right) ,\eta \right\rangle \right) , \end{aligned}$$

(4.9)

$$\begin{aligned}&g^{ij}\left( \left\langle D_\frac{\partial }{\partial t}\left( \left( x_{t}\right) _{*}\nabla _{\frac{\partial }{\partial u^j}}^{t}\frac{\partial }{\partial u^i}\right) |_{t=0},\eta \right\rangle \right) =g^{ij}\left\langle D_{\nabla _{\frac{\partial }{\partial u^j}}\frac{\partial }{\partial u^i}}^\bot \eta ,\eta \right\rangle . \end{aligned}$$

(4.10)

Hence,

$$\begin{aligned} \left\langle D_\frac{\partial }{\partial t}H_{t}|_{t=0},\eta \right\rangle&=\left\langle g^{ij}\left( D_{\frac{\partial }{\partial u^i}}^\bot D_\frac{\partial }{\partial u^j}^\bot \eta -D_{\nabla _{\frac{\partial }{\partial u^i}}\frac{\partial }{\partial u^j}}^\bot \eta \right) ,\eta \right\rangle \\&\quad +g^{ik}g^{jl}\left\langle h\left( \frac{\partial }{\partial u^k},\frac{\partial }{\partial u^l}\right) ,\eta \right\rangle \left\langle h\left( \frac{\partial }{\partial u^i},\frac{\partial }{\partial u^j}\right) ,\eta \right\rangle \\&=\left\langle \triangle _{M}^\bot \eta ,\eta \right\rangle +g^{ik}g^{jl}\left\langle h\left( \frac{\partial }{\partial u^k},\frac{\partial }{\partial u^l}\right) ,\eta \right\rangle \left\langle h\left( \frac{\partial }{\partial u^i},\frac{\partial }{\partial u^j}\right) ,\eta \right\rangle \\&=\left\langle \triangle _{M}^\bot \eta +g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl} ,\eta \right\rangle , \end{aligned}$$

where $h_{ij}=h(\frac{\partial }{\partial u^i},\frac{\partial }{\partial u^j})$. Furthermore,

$$\begin{aligned}&\left\langle D_\frac{\partial }{\partial t} (x_{t}^\bot -\xi )|_{t=0},\eta \right\rangle =\left\langle D_\frac{\partial }{\partial t} x_{t}|_{t=0}- D_\frac{\partial }{\partial t}(x_t)^\top |_{t=0},\eta \right\rangle \\&\quad =\left\langle \eta ,\eta \right\rangle -\left\langle D_\frac{\partial }{\partial t} \left( (g_t)^{ij}\left\langle x_{t},\frac{\partial F}{\partial u^i}\right\rangle \frac{\partial F}{\partial u^j}\right) |_{t=0},\eta \right\rangle \\&\quad =\left\langle \eta ,\eta \right\rangle -\left\langle D_{x^\top }\eta ,\eta \right\rangle =\left\langle \eta ,\eta \right\rangle -\left\langle D^\bot _{x^\top }\eta ,\eta \right\rangle . \end{aligned}$$

Therefore,

$$\begin{aligned} \left\langle D_{\frac{\partial }{\partial t}}(H_{t}+x_{t}^\bot -\xi ),\frac{\partial F}{\partial t}\right\rangle |_{t=0}=\left\langle \triangle _{M}^\bot \eta -D^\bot _{x^\top }\eta +g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl} +\eta ,\eta \right\rangle \end{aligned}$$

Meanwhile,

$$\begin{aligned} \left\langle D_{\frac{\partial }{\partial t}}(\nabla ^t\left\langle x_t,\xi \right\rangle )|_{t=0},\eta \right\rangle&=\left\langle (g_t)^{ij}\frac{\partial }{\partial u^i}\left\langle x_t,\xi \right\rangle D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial u^i}|_{t=0},\eta \right\rangle \\&=\left\langle g^{ij}\frac{\partial }{\partial u^i}\left\langle x,\xi \right\rangle D_{\frac{\partial }{\partial u^i}}\eta ,\eta \right\rangle =\left\langle D^\bot _{\nabla \left\langle x,\xi \right\rangle }\eta ,\eta \right\rangle \\&=-\left\langle D^\bot _{A_\xi (x^\top )}\eta ,\eta \right\rangle \end{aligned}$$

since $\xi$ is parallel along x.

By summing up, we have proved the following second variation formulas for $\xi$-submanifolds:

Theorem 4.3

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a $\xi$-submanifold. Then for any compact normal variation $F:M^m\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$, we have

$$\begin{aligned} V_{\xi }^{''}(0)&=-\int _M\left( \left\langle \triangle _{M}^\bot (\eta )- D^\bot _{x^\top +A_\xi (x^\top )}\eta +g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl}+\eta ,\eta \right\rangle \right. \nonumber \\&\quad \left. +\left\langle \nabla \left\langle x,\xi \right\rangle , D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial t}|_{t=0}\right\rangle \right) e^{-f}\mathrm{d}V, \end{aligned}$$

(4.11)

$$\begin{aligned} {{\bar{V}}}_\xi ^{''}(0)&=-\int _M\left( \left\langle \triangle _{M}^\bot (\eta )- D^\bot _{x^\top +A_\xi (x^\top )}\eta +g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl} +\eta ,\eta \right\rangle \right. \nonumber \\&\quad \left. +\left\langle \nabla \left\langle x,\xi \right\rangle , D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial t}|_{t=0}\right\rangle \right) e^{-{{\bar{f}}}}\mathrm{d}V. \end{aligned}$$

(4.12)

In order to simplify the second variation formulas, we introduce the following definition:

Definition 4.2

(SN-variation). A variation $F:M^m\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ of an immersion $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is called specially normal (or simply SN) if it is normal and $\frac{\partial ^2F}{\partial t^2}|_{t=0}=0$.

Remark 4.2

The introduction of the SN-variation is based on the observation that the Hessian $\mathrm{Hess}(f)$ at a given point p of a smooth function f on a Riemannian manifold $\tilde{N}$, $p\in \tilde{N}$, is determined only by those local values of f along the simplest curves $\tilde{\gamma }$ passing through the point p. For example, if we choose $\tilde{\gamma }$ to be geodesic ones, then the second derivatives can be computed as

$$\begin{aligned} \left. \frac{d^2}{dt^2}\right| _{t=0}(f(\tilde{\gamma }))=\mathrm{Hess}(f)(\tilde{\gamma }'(0),\tilde{\gamma }'(0)), \end{aligned}$$

implying that f is (semi-)convex at p if and only if $\left. \frac{d^2}{dt^2}\right| _{t=0}(f(\tilde{\gamma }))\ge 0$ for all of these geodesics $\tilde{\gamma }$.

Clearly, for any $\eta \in \Gamma (T^\bot M)$, SN-variations with variation vector field $\eta$ always exist in our present case. For example, we can choose

$$\begin{aligned} F(p,t)=x(p)+\psi (t)\eta (p),\quad \forall \, (p,t)\in M^m\times (-\varepsilon ,\varepsilon ) \end{aligned}$$

where $\psi$ is any smooth function satisfying $\psi (0)=\psi ''(0)=0$, $\psi '(0)=1$.

Corollary 4.4

(The simplified second variation formulas). Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a $\xi$-submanifold. Then for any compact SN-variation $F:M^m\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ it holds that

$$\begin{aligned} V_{\xi }^{''}(0)&=-\int _M\Big (\left\langle (\triangle _{M}^\bot -D^\bot _{x^\top +A_\xi (x^\top )}+1)\eta +g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl} ,\eta \right\rangle \Big )e^{-f}\mathrm{d}V, \end{aligned}$$

(4.13)

$$\begin{aligned} {{\bar{V}}}_\xi ^{''}(0)&=-\int _M\Big (\left\langle (\triangle _{M}^\bot -D^\bot _{x^\top +A_\xi (x^\top )}+1)\eta +g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl} ,\eta \right\rangle \Big )e^{-{{\bar{f}}}}\mathrm{d}V. \end{aligned}$$

(4.14)

Remark 4.3

From the above discussion, one may naturally think of the variational characterization of the usual submanifolds with parallel mean curvature vector in the Euclidean space. In fact, our computations and argument in this section essentially apply to this situation. For example, a suitable functional $\tilde{V}_\xi$ may be defined by

$$\begin{aligned} \tilde{V}_\xi =\int _M e^{\left\langle x,\xi \right\rangle }\mathrm{d}V_x,\quad \forall x\in {{\mathcal {M}}} \end{aligned}$$

and the first variation formula of $\tilde{V}_\xi$ is given in the following

Proposition 4.5

Let $x\in {{\mathcal {M}}}$ be fixed and $\xi :M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a smooth map. Suppose that F is a compact variation of x. Then

$$\begin{aligned} \tilde{V}_\xi '(t)=-\int _M \left\langle (H_{t}-\xi )+\nabla ^t\left\langle x_t,\xi \right\rangle ,\frac{\partial F}{\partial t}\right\rangle e^{\left\langle x,\xi \right\rangle }\mathrm{d}V_t. \end{aligned}$$

(4.15)

In particular, if F is a normal variation of x, then

$$\begin{aligned} \tilde{V}_\xi '(0)=-\int _M\left\langle H-\xi ,\eta \right\rangle e^{\left\langle x,\xi \right\rangle }\mathrm{d}V. \end{aligned}$$

(4.16)

Corollary 4.6

An immersion $x\in {{\mathcal {M}}}$ has a parallel mean curvature vector if and only if there exists a parallel normal vector field $\xi \in \Gamma (T^\bot M)$ such that x is the critical point of the functional $\tilde{V}_\xi$ for all the compact normal variations of x.

Accordingly, the second variation formula for a submanifold $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ with parallel mean curvature vector $H\equiv \xi$ may be described as

Theorem 4.7

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be an immersed submanifold with parallel mean curvature H. Then for any compact normal variation $F:M^m\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ we have

$$\begin{aligned} \tilde{V}_H^{''}(0)=-\int _M\left( \left\langle \triangle _{M}^\bot (\eta )+D^\bot _{\nabla \left\langle x,H\right\rangle } \eta ,\eta \right\rangle +|A_\eta |^2+ \left\langle \nabla \left\langle x,H\right\rangle , D_{\frac{\partial }{\partial t}}\frac{\partial F}{\partial t}|_{t=0}\right\rangle \right) e^{\left\langle x,H\right\rangle }\mathrm{d}V.\nonumber \\ \end{aligned}$$

(4.17)

5 The instabilities of the typical examples

The most natural stability definition to the functional $V_\xi$ is as follows:

Definition 5.1

A $\xi$-submanifold $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is called stable if $V_\xi (x)<+\infty$ and for every compact SN-variation $F:M^m\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ of x it holds that $V_\xi ''(0)\ge 0$ or, equivalently, ${{\bar{V}}}_\xi ''(0)\ge 0$.

In this section, we shall show that, as $\xi$-submanifolds, all the typical examples given in Sect. 2 are not stable in the sense of Definition 5.1.

Write the second fundamental form h of x locally as $h=h_{ij}\omega ^i\omega ^j=h^\alpha _{ij}e_\alpha$ with respect to an orthonormal tangent frame field $\{e_i;\ 1\le i\le m\}$ with dual $\{\omega ^i\}$ and an orthonormal normal frame field $\{e_\alpha ;\ m+1\le \alpha \le m+p\}$, and denote

$$\begin{aligned} {{\mathcal {L}}}=\triangle ^\bot _{M^m}-D^\bot _{x^\top +A_\xi (x^\top )},\quad L={{\mathcal {L}}}+\left\langle h_{ij},\cdot \right\rangle h_{ij} +1,\quad \tilde{{\mathcal {L}}}=\triangle _{M^m} -\nabla _{x^\top +A_\xi (x^\top )},\qquad \end{aligned}$$

(5.1)

where $\triangle ^\bot _{M^m}$, $\triangle _{M^m}$ are Laplacians on $T^\bot M^m$, $TM^m$, respectively, and sometimes we shall omit the subscript “$_{M^m}$” if no confusion is made. It follows that

$$\begin{aligned} Q(\eta ,\eta ):\equiv -\int _M\left\langle L(\eta ) ,\eta \right\rangle e^{-f}\mathrm{d}V, \end{aligned}$$

(5.2)

and that, for any parallel normal vector field N,

$$\begin{aligned} L(N)=N+\left\langle h_{ij},N\right\rangle h_{ij}. \end{aligned}$$

(5.3)

Lemma 5.1

$$\begin{aligned} L(\phi \eta )=(\tilde{{\mathcal {L}}}\phi )\eta +\phi L(\eta )+2D^\bot _{\nabla \phi }\eta ,\quad \phi \in C^\infty (M^m),\, \eta \in \Gamma (T^\bot M^m). \end{aligned}$$

(5.4)

Proof

We compute directly

$$\begin{aligned} L(\phi \eta )&=\triangle ^\bot (\phi \eta )-D^\bot _{x^\top +A_\xi (x^\top )}(\phi \eta )+\left\langle h_{ij},\phi \eta \right\rangle h_{ij}+\phi \eta \\&=(\triangle \phi )\eta +2D^\bot _{\nabla \phi }\eta +\phi \triangle ^\bot \eta -(\nabla _{x^\top +A_\xi (x^\top )}\phi )\eta \\&\quad -\phi (D^\bot _{x^\top +A_\xi (x^\top )}\eta )+\phi \left\langle h_{ij},\eta \right\rangle h_{ij}+\phi \eta \\&=(\triangle -\nabla _{x^\top +A_\xi (x^\top )})\phi \eta +\phi (\triangle ^\bot -D^\bot _{x^\top +A_\xi (x^\top )}+\left\langle h_{ij},\cdot \right\rangle h_{ij}+1)\eta +2D^\bot _{\nabla \phi }\eta \\&=(\tilde{{\mathcal {L}}}\phi )\eta +\phi (L\eta )+2D^\bot _{\nabla \phi }\eta . \end{aligned}$$

$\square$

Lemma 5.2

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a $\xi$-submanifold. Then for any $\eta _1,\eta _2\in \Gamma (T^\bot M^m)$ one of which is compactly supported, it holds that

$$\begin{aligned} \int _M\left\langle \eta _1,{{\mathcal {L}}}\eta _2\right\rangle e^{-f}\mathrm{d}V=-\int _M\left\langle D^\bot \eta _1,D^\bot \eta _2\right\rangle e^{-f}\mathrm{d}V. \end{aligned}$$

(5.5)

Similarly, for any $\phi _1, \phi _2\in C^\infty (M^m)$ one of which is compactly supported, it holds that

$$\begin{aligned} \int _M\phi _1\tilde{{\mathcal {L}}}\phi _2 e^{-f}\mathrm{d}V=-\int _M\left\langle \nabla \phi _1,\nabla \phi _2\right\rangle e^{-f}\mathrm{d}V. \end{aligned}$$

(5.6)

Proof

To prove the two formulas, it suffices to use the Divergence Theorem and the following equalities:

$$\begin{aligned}&\left\langle \eta _1,{{\mathcal {L}}}\eta _2\right\rangle e^{-f}={\mathrm{div}}\left( \left\langle \eta _1,D^\bot _{e_i}\eta _2\right\rangle e^{-f}e_i\right) -\left\langle D^\bot \eta _1,D^\bot \eta _2\right\rangle e^{-f}, \end{aligned}$$

(5.7)

$$\begin{aligned}&\phi _1\tilde{{\mathcal {L}}}\phi _2e^{-f}={\mathrm{div}}\left( \phi _1\nabla _{e_i}\phi _2e^{-f}e_i\right) -\left\langle \nabla \phi _1,\nabla \phi _2\right\rangle e^{-f}. \end{aligned}$$

(5.8)

$\square$

Lemma 5.3

For any $\phi \in C^\infty _0(M^m)$ and $\eta \in \Gamma (T^\bot M^m)$, it holds that

$$\begin{aligned} \int _M \left\langle \phi \eta ,L(\phi \eta )\right\rangle e^{-f}\mathrm{d}V=\int _M\phi ^2 \left\langle \eta ,L(\eta )\right\rangle e^{-f}\mathrm{d}V-\int _M|\nabla \phi |^2 |\eta |^2e^{-f}\mathrm{d}V. \end{aligned}$$

(5.9)

Proof

By (5.4) and (5.6), we find

$$\begin{aligned}&\int _M \left\langle \phi \eta ,L(\phi \eta )\right\rangle e^{-f}\mathrm{d}V=\int _M \left\langle \phi \eta ,(\tilde{{\mathcal {L}}}\phi )\eta +\phi L\eta +2D^\bot _{\nabla \phi }\eta \right\rangle e^{-f}\mathrm{d}V\\&\quad =\int _M (\phi |\eta |^2)\tilde{{\mathcal {L}}}\phi e^{-f}\mathrm{d}V+\int _M\phi ^2\left\langle \eta ,L\eta \right\rangle e^{-f}\mathrm{d}V+\int _M\left\langle \eta ,D^\bot _{\nabla \phi ^2}\eta \right\rangle e^{-f}\mathrm{d}V\\&\quad =-\int _M((|\nabla \phi |^2|\eta |^2)+\frac{1}{2}\left\langle \nabla \phi ^2,\nabla |\eta |^2\right\rangle ) e^{-f}\mathrm{d}V+\int _M\phi ^2\left\langle \eta ,L\eta \right\rangle e^{-f}\mathrm{d}V\\&\qquad +\frac{1}{2}\int _M\nabla _{\nabla \phi ^2}|\eta |^2 e^{-f}\mathrm{d}V\\&\quad =\int _M\phi ^2\left\langle \eta ,L\eta \right\rangle e^{-f}\mathrm{d}V-\int _M |\nabla \phi |^2|\eta |^2 e^{-f}\mathrm{d}V. \end{aligned}$$

$\square$

Proposition 5.4

As $\xi$-submanifolds, all m-planes in ${{\mathbb {R}}}^{m+p}$ are not stable.

Proof

For an m-plane $x:P^m\subset {{\mathbb {R}}}^{m+p}$, let o be the orthogonal projection on $P^m$ of the origin O. Then $\xi ={\mathop {Oo}\limits ^{{\rightarrow }}}$. Denote by $B_R(o)\subset P$ the closed ball of radius $R>0$ centered at the fixed point o:

$$\begin{aligned} B_R(o)=\{x\in P;\ |x^\top |\equiv |x-\xi |\le R\}. \end{aligned}$$

Let N be a unit constant vector in ${{\mathbb {R}}}^m$ orthogonal to $P^m$ and $\phi _R$ be a cut-off function on $P^m$ satisfying

$$\begin{aligned} (\phi _R)|_{B_R(o)}\equiv 1,\quad (\phi _R)|_{P^m\backslash B_{R+2}(o)}\equiv 0,\quad |\nabla \phi |\le 1,\quad R>0. \end{aligned}$$

Define $\eta _R=\phi _RN$. Then $\eta _R$ is compactly supported and can be chosen to be a variation vector field for some SN-variation. By (5.9) and (5.3),

$$\begin{aligned} Q(\eta _R,\eta _R)&=-\int _M \left\langle \phi _R N,L(\phi _R N)\right\rangle e^{-f}\mathrm{d}V \\&=-\int _{P^m}\phi _R^2 \left\langle N,L(N)\right\rangle e^{-f}\mathrm{d}V+\int _{P^m}|\nabla \phi _R|^2 e^{-f}\mathrm{d}V\\&=-\int _{P^m}\phi _R^2 \left\langle N,N+\left\langle h_{ij},N\right\rangle h_{ij}\right\rangle e^{-f}\mathrm{d}V+\int _{P^m}|\nabla \phi _R|^2 e^{-f}\mathrm{d}V\\&\le -\int _{P^m}\phi _R^2 e^{-f}\mathrm{d}V+\int _{B_{R+2}(o)\backslash B_R(o)} e^{-f}\mathrm{d}V \ \rightarrow -\int _{P^m} e^{-f}\mathrm{d}V<0 \end{aligned}$$

when $R\,\rightarrow +\infty$ since $\int _{P^m} e^{-f}\mathrm{d}V<+\infty$. Thus for large R, we have $Q(\eta _R,\eta _R)<0$. $\square$

Proposition 5.5

As $\xi$-submanifolds, the standard m-spheres $S^m(r)$ are all non-stable.

Proof

For the standard sphere $S^m(r)\subset {{\mathbb {R}}}^{m+1}\subset {{\mathbb {R}}}^{m+p}$, we have $h=-\frac{1}{r^2}g\,x$, $x^\bot =x$ and $\xi =\left( -\frac{m}{r^2}+1\right) x$. Choose the variation vector field $\eta =x$ so that ${{\mathcal {L}}}\eta =0$. It follows that

$$\begin{aligned} Q(\eta ,\eta )&\le -\int _{S^m(r)}\left\langle \eta ,L(\eta )\right\rangle e^{-f}\mathrm{d}V_{S^m(r)}=-\int _{S^m(r)}\left( \sum \left\langle h_{ij},\eta \right\rangle ^2+|x|^2\right) e^{-f}\mathrm{d}V_{S^m(r)}\\&=-(m+r^2)\int _{S^m(r)}e^{-f}\mathrm{d}V_{S^m(r)}<0. \end{aligned}$$

$\square$

From Proposition 5.4 and Proposition 5.5, we easily find

Corollary 5.6

The product $\xi$-submanifolds $S^{m_1}(r_1)\times \cdots \times S^{m_k}(r_k)\times P^{n_1}\times \cdots \times P^{n_l}$ are not stable.

A more general conclusion than Proposition 5.5 is the following

Proposition 5.7

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a compact $\xi$-submanifold. If x has a non-trivial parallel normal vector field, then x is not stable. In particular, all compact $\lambda$-hypersurfaces and compact $\xi$-submanifold with $\xi \ne 0$ are not stable.

Proof

Let $\eta \ne 0$ be a parallel normal vector field. Then $\eta$ can be chosen to be the variation vector field of some SN-variation F of x. Since $\triangle ^\bot \eta =D^\bot _{x^\top +A_\xi (x^\top )}\eta =0$, it then follows from (4.13) that

$$\begin{aligned} Q(\eta ,\eta )=-\int _M\left( \sum \left\langle h_{ij},\eta \right\rangle ^2+|\eta |^2\right) e^{-f}\mathrm{d}V<0. \end{aligned}$$

$\square$

Corollary 5.8

Any compact and simply connected $\xi$-submanifold with flat normal bundle is not stable.

To end this section, we would like to remark that, by using suitably chosen cut-off functions, say, the cut-off functions $\phi _R$ introduced in Sect. 7 for large enough numbers $R>0$, we can extend the above instability conclusions to more general complete case. For example, the following conclusion is also true:

Theorem 5.9

Any complete and properly immersed $\xi$-submanifold with a non-trivial parallel normal vector field $\eta$ is not stable.

Proof

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a complete and properly immersed $\xi$-submanifold and N be a non-trivial parallel normal vector field of x. Without loss of generality, we assume that $\int _Me^{-f}\mathrm{d}V<\infty$ and $|N|^2=1$. For a Large $R>0$, define $\eta _R:=\phi _RN$. Choose an SN-variation of x with $\eta _R$ being its variation vector field. Then, by (5.2), (5.3) and Lemma 5.3, we have

$$\begin{aligned} Q(\eta _R,\eta _R)&=-\int _M \left\langle \phi _R N,L(\phi _R N)\right\rangle e^{-f}\mathrm{d}V \\&=-\int _M\phi _R^2 \left\langle N,L(N)\right\rangle e^{-f}\mathrm{d}V+\int _M|\nabla \phi _R|^2 e^{-f}\mathrm{d}V\\&=-\int _{M}\phi _R^2 \left\langle N,N+\left\langle h_{ij},N\right\rangle h_{ij}\right\rangle e^{-f}\mathrm{d}V+\int _{B_{2R}(o)\backslash B_R(o)}|\nabla \phi _R|^2 e^{-f}\mathrm{d}V\\&\le -\int _{M}\phi _R^2 e^{-f}\mathrm{d}V+\int _{M\backslash B_R(o)} e^{-f}\mathrm{d}V \ \rightarrow -\int _{M} e^{-f}\mathrm{d}V<0\quad (R\rightarrow +\infty ), \end{aligned}$$

since $\lim _{R\rightarrow +\infty }\int _{M\backslash B_R(o)} e^{-f}\mathrm{d}V=0$. So that there is an R large enough such that we have $Q(\eta _R,\eta _R)<0$. $\square$

Corollary 5.10

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a complete and properly immersed $\xi$-submanifold. Then, x is not stable if any of the following three holds:

• (1) the codimension $p=1$;

• (2) $p\ge 2$ and $\xi \ne 0$;

• (3) $M^m$ is simply connected and the normal bundle of x is flat.

Remark 5.1

Up to now, it is still unclear for the existence of stable $\xi$-submanifolds in the sense of Definition 5.1 . Other stability problems have been previously discussed for both self-shrinker hypersurfaces and $\lambda$-hypersurfaces. For example, Colding and Minicozzi introduced a notion of ${{\mathcal {F}}}$-functional and proved that self-shrinkers are exactly critical points of the ${{\mathcal {F}}}$-functional ([13]). They also proved that the standard sphere and hyperplane are the only two complete ${{\mathcal {F}}}$-stable hypersurface self-shrinkers of polynomial volume growth. Furthermore, in [9], Cheng and Wei extended the above ${{\mathcal {F}}}$-functional to $\lambda$-hypersurfaces and studied the corresponding ${{\mathcal {F}}}$-stability. In particular, they proved that the standard sphere ${{\mathbb {S}}}^m(r)$ of radius r is ${{\mathcal {F}}}$-unstable as a $\lambda$-hypersurface if and only if $\sqrt{m}< r\le \sqrt{m+1}$.

6 The $W_\xi$-stability of $\xi$-submanifolds

By the discussion of last section, it turns out that the concept of stability given in Definition 5.1 is over-strong in a sense. So it is natural and interesting to find a suitably weaker stability definition for $\xi$-submanifolds. Motivated by the “weighted-volume-preserving” variations of hypersurfaces (see [27]), we can introduce the $W_\xi$-stability in the following way.

Note that, by [27], a compact variation F of a hypersurface $x:M^m\rightarrow {{\mathbb {R}}}^{m+1}$ is called “weighted-volume-preserving” if $\int _M\left\langle \left. \frac{\partial }{\partial F}\right| _{t=0},n\right\rangle e^{-\frac{1}{2}|x|^2}=0$ where n is the unit normal vector field. Since a normal vector field $N=\lambda n$ is parallel if and only if $\lambda ={\mathrm{const}}$, it follows that F is “weighted-volume-preserving” if and only if $\int _M\left\langle \left. \frac{\partial }{\partial F}\right| _{t=0},N\right\rangle e^{-\frac{1}{2}|x|^2}\mathrm{d}V=0$ for all parallel normal vector field N. This recommends us to make the following generalization:

Definition 6.1

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be an immersion. A compact SN-variation $F:M^m\times (-\varepsilon ,\varepsilon )\rightarrow {{\mathbb {R}}}^{m+p}$ of x is called VP (“weighted-volume-preserving”) if the corresponding variation vector $\eta \equiv \left. \frac{\partial F}{\partial t}\right| _{t=0}$ satisfies

$$\begin{aligned} \int _M\left\langle \eta ,N\right\rangle e^{-f}\mathrm{d}V=0,\quad \forall \, N\in \Gamma (T^\bot M)\text { satisfying } D^\bot N\equiv 0. \end{aligned}$$

(6.1)

Remark 6.1

It is clear that, in the special case of codimension 1, VP-variations defined here are nothing but the “weighted-volume-preserving” ones that were considered in [27].

Definition 6.2

A $\xi$-submanifold $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is called $W_\xi$-stable if $V_\xi (x)<+\infty$ and for every VP-variation it holds that $V_\xi ''(0)\ge 0$.

Then, we have

Theorem 6.1

Any of the m-planes is $W_\xi$-stable.

Proof

For an m-plane $x:P^m\subset {{\mathbb {R}}}^{m+p}$, let $\eta$ be an arbitrary normal vector field on $P^m$ with compact support. Then, we have $A_\eta \equiv 0$, $x-\xi =x^\top$ and

$$\begin{aligned} L=\triangle _{P^m}^\bot -D^\bot _{x^\top }+1. \end{aligned}$$

Clearly, there are constant normal basis $e_\alpha$, $\alpha =m+1,\ldots ,m+p$. So $\eta$ can be expressed by $\eta =\sum \eta ^\alpha e_\alpha$ with $\eta ^\alpha \in C^\infty _0(P^m)$. Consequently,

$$\begin{aligned} L(\eta )=\sum \tilde{L}(\eta ^\alpha )e_\alpha ,\quad \left\langle L\eta ,\eta \right\rangle =\sum \eta ^\alpha \tilde{L}\eta ^\alpha , \end{aligned}$$

where $\tilde{L}=\triangle _{P^m}-\nabla _{x^\top }+1$. Now we make the following

Claim:The eigenvalues of the operator$-\tilde{L}$are$\lambda _n=n-1$with$n=0,1,\ldots$.

To prove this claim, we need to make use of the multivariable Hermitian polynomials${{\mathcal {H}}}_{n_1\cdots n_m}$ on ${{\mathbb {R}}}^m$, labelled with $0\le n_1,\ldots , n_m<+\infty$, which are defined by the expansion (see [14] and [15] for the detail)

$$\begin{aligned}&e^{-\frac{|u-t|^2}{2}}=e^{-\frac{|u|^2}{2}}\sum _{n_1,\ldots , n_m}\frac{(t^1)^{n_1}\cdots (t^m)^{n_m}}{n_1!\cdots n_m!}{{\mathcal {H}}}_{n_1\cdots n_m}(u), \nonumber \\&\quad u=(u^1,\ldots ,u^m),\ t=(t^1,\ldots t^m)\in {{\mathbb {R}}}^m, \end{aligned}$$

(6.2)

or equivalently

$$\begin{aligned}&e^{-\frac{|t|^2}{2}+\left\langle t,u\right\rangle }=\sum _{n_1,\ldots , n_m}\frac{(t^1)^{n_1}\cdots (t^m)^{n_m}}{n_1!\cdots n_m!}{{\mathcal {H}}}_{n_1\cdots n_m}(u),\nonumber \\&\quad u=(u^1,\ldots ,u^m),\ t=(t^1,\ldots t^m)\in {{\mathbb {R}}}^m. \end{aligned}$$

(6.3)

It is clear that

$$\begin{aligned} {{\mathcal {H}}}_{n_1\cdots n_m}(u)={{\mathcal {H}}}_{n_1}(u^1)\,\ldots \,{{\mathcal {H}}}_{n_m}(u^m),\quad \forall u=(u^1,\ldots ,u^m)\in {{\mathbb {R}}}^m \end{aligned}$$

(6.4)

where, for each $i=1,\ldots ,m$, ${{\mathcal {H}}}_{n_i}(u^i)$ is the Hermitian Polynomial of one variable $u^i$ defined by

$$\begin{aligned} e^{-\frac{1}{2}|t^i|^2+u^it^i}=\sum _{n_i}\frac{(t^i)^{n_i}}{n_i!}{{\mathcal {H}}}_{n_i}(u^i),\quad u^i,t^i\in {{\mathbb {R}}}. \end{aligned}$$

(6.5)

By (6.5), we easily find that

$$\begin{aligned} {{\mathcal {H}}}_{n_i+1}(u^i)=u^i{{\mathcal {H}}}_{n_i}-n_i{{\mathcal {H}}}_{n_i-1},\quad \frac{\mathrm{d} }{\mathrm{d} u^i}{{\mathcal {H}}}_{n_i}(u^i)=n_i{{\mathcal {H}}}_{n_i-1},\ i=1,\ldots ,m \end{aligned}$$

(6.6)

implying that

$$\begin{aligned} \left( -\frac{d^2}{(du^i)^2}+u^i\frac{\mathrm{d} }{\mathrm{d} u^i}\right) {{\mathcal {H}}}_{n_i}(u^i)=n_i{{\mathcal {H}}}_{n_i}(u^i),\quad i=1,\ldots ,m. \end{aligned}$$

(6.7)

Consequently, by (6.4), we have

$$\begin{aligned} \left( -\triangle _{{{\mathbb {R}}}^m}+\nabla _{u}\right) {{\mathcal {H}}}_{n_1\cdots n_m}(u)=\left( \sum _{i=1}^mn_i\right) {{\mathcal {H}}}_{n_1\cdots n_m}(u),\quad \forall n_1,\ldots ,n_m\ge 0. \end{aligned}$$

(6.8)

It is known that all these multivariable Hermitian polynomials are weighted square integrable with the weight $e^{-\frac{|u|^2}{2}}$, that is

$$\begin{aligned} {{\mathcal {H}}}_{n_1\cdots n_m}\in L^2_w({{\mathbb {R}}}^m):=\overline{\{\varphi \in C^\infty ({{\mathbb {R}}}^m);\ \int _{{{\mathbb {R}}}^m}\varphi ^2 e^{-f}\mathrm{d}V_{{{\mathbb {R}}}^m}<+\infty \}}. \end{aligned}$$

Consequently, integers $\sum _{i=1}^m n_i$, for all $n_1,\ldots ,n_m\ge 0$, are eigenvalues of the operator $-\triangle _{{{\mathbb {R}}}^m}+\nabla _{u}$ acting on $L^2_w({{\mathbb {R}}}^m)$. By making a change of coordinates on ${{\mathbb {R}}}^{m+p}$ we can assume $x^i-\xi ^i=u^i$, $i=1,2,\ldots , m$, for $x\in P^m$. Thus, (6.8) shows that $-\tilde{L}+1$ has $n=0,1,\ldots$ as its eigenvalues, or equivalently, $n-1=-1,0,1,\ldots$ are eigenvalues of $-\tilde{L}$ where constants are those eigenfunctions corresponding to $-1$.

To complete the claim, we also have to show that $\{{{\mathcal {H}}}_{n_1\cdots n_m};\ n_1,\ldots ,n_m\ge 0\}$ is a complete basis for the space of smooth and weighted square integrable functions on ${{\mathbb {R}}}^m$. For doing this, we let E be the orthogonal complement in $L^2_w({{\mathbb {R}}}^m)$ of the closure of the linear span of all ${{\mathcal {H}}}_{n_1\ldots n_m}$, that is,

$$\begin{aligned} E:=(\overline{{\mathrm{Span\,}}\{{{\mathcal {H}}}_{n_1\ldots n_m},\ n_1,\ldots , n_m=0,1,\ldots \}})^\bot . \end{aligned}$$

For any $\varphi \in E$, we have

$$\begin{aligned} 0=(\varphi ,{{\mathcal {H}}}_{n_1\ldots n_m})_w:=\int _{{{\mathbb {R}}}^m}\varphi (u){{\mathcal {H}}}_{n_1\ldots n_m}(u)e^{-f}\mathrm{d}V_{{{\mathbb {R}}}^m},\quad n_1,\ldots , n_m=0,1,\ldots . \end{aligned}$$

It then easily follows from (6.3) that ${{\mathcal {F}}}(\varphi e^{-f})=0$ where ${{\mathcal {F}}}$ is the usual multivariable Fourier transformation. Since ${{\mathcal {F}}}$ is injective, we obtain that $\varphi e^{-f}=0$ implying $\varphi \equiv 0$. This shows that $E=0$ and thus

$$\begin{aligned} L^2_w({{\mathbb {R}}}^m)=\overline{{\mathrm{Span\,}}\{{{\mathcal {H}}}_{n_1\cdots n_m},\ n_1,\ldots , n_m=0,1,\ldots \}}. \end{aligned}$$

(6.9)

Now suppose $\eta =\sum \eta ^\alpha e_\alpha$ is a compact normal vector field that can be taken as a VP-variation vector field. Then for each $\alpha$, we have

$$\begin{aligned} \eta ^\alpha \in S^{\infty ,2}_w(P^m):=\left\{ \varphi \in C^\infty (P^m);\ \int _{P^m}\varphi ^2 e^{-f}\mathrm{d}V_{P^m}<+\infty \right\} . \end{aligned}$$

Since $\tilde{L}$ is self-adjoint with respect to the weighted measure $e^{-f}\mathrm{d}V$, we know that it is diagonalizable, that is, any compactly supported smooth function can be decomposed into a sum of some eigenfunctions of $\tilde{L}$. In particular, we can write for each $\alpha =m+1,\ldots ,m+p$,

$$\begin{aligned} \eta ^\alpha =\eta ^\alpha _0+\sum _{k\ge 1}\eta ^\alpha _k,\quad \eta ^\alpha _0\in {{\mathbb {R}}}, \end{aligned}$$

(6.10)

where $\eta ^\alpha _k\in S^{\infty ,2}_{w}(P^m)$ satisfying $\tilde{L}(\eta ^\alpha _k)=-\lambda _k\eta ^\alpha _k$, $k\ge 0$. Furthermore, the self-adjointness of $\tilde{L}$ also implies that, for each pair of $k\ne l$, $\eta ^\alpha _k$ and $\eta ^\alpha _l$ are orthogonal, that is

$$\begin{aligned} \int _{P^m}\sum _\alpha \eta ^\alpha _k\eta ^\alpha _le^{-f}\mathrm{d}V=0,\quad k\ne l. \end{aligned}$$

(6.11)

Since $\eta$ is a VP-variation vector field, we have by (6.10) and (6.1) that $\int _{P^m}\eta ^\alpha e^{-f}\mathrm{d}V=0$ for all $\alpha =m+1,\ldots ,m+p$. It then follows from (6.11) that $\eta ^\alpha _0=0$, $\alpha =m+1,\ldots ,m+p$. Therefore,

$$\begin{aligned} \int _{P^m}\sum _\alpha |\eta ^\alpha |^2e^{-f}\mathrm{d}V=\int _{P^m}\sum _\alpha \sum _{k,l\ge 1}\eta ^\alpha _k\eta ^\alpha _le^{-f}\mathrm{d}V=\sum _\alpha \sum _{k\ge 1}\int _{P^m}|\eta ^\alpha _k|^2e^{-f}\mathrm{d}V. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \int _{P^m}\sum _\alpha \eta ^\alpha (-\tilde{L}\eta ^\alpha )e^{-f}\mathrm{d}V&=\int _{P^m}\sum _\alpha \sum _{k\ge 1}\eta ^\alpha _k\sum _{l\ge 1}(-\tilde{L}\eta ^\alpha _l)e^{-f}\mathrm{d}V\\&=\sum _\alpha \sum _{k,l\ge 1}\int _{P^m}\lambda _l\eta ^\alpha _k\eta ^\alpha _le^{-f}\mathrm{d}V =\sum _\alpha \sum _{k\ge 1}\lambda _k\int _{P^m}|\eta ^\alpha _k|^2e^{-f}\mathrm{d}V\\&\ge \lambda _1\sum _\alpha \sum _k\int _{P^m}|\eta ^\alpha _k|^2e^{-f}\mathrm{d}V =\lambda _1\sum _\alpha \int _{P^m}|\eta ^\alpha |^2e^{-f}\mathrm{d}V\ge 0 \end{aligned}$$

implying that

$$\begin{aligned} Q(\eta ,\eta )&=-\int _{P^m}\left\langle \eta ,L\eta \right\rangle e^{-f}\mathrm{d}V=\int _{P^m}\sum _\alpha \eta ^\alpha (-\tilde{L}\eta ^\alpha )e^{-f}\mathrm{d}V\\&= \sum _\alpha \int _{P^m}\eta ^\alpha (-\tilde{L}\eta ^\alpha )e^{-f}\mathrm{d}V\ge 0. \end{aligned}$$

$\square$

Theorem 6.2

As a $\xi$-submanifold, the index ${\mathrm{ind}}(S^m(r))$ of the standard m-sphere $S^m(r)$ with respect to VP-variations is no less than $m+1$. Furthermore, ${\mathrm{ind}}(S^m(r))=m+1$ if and only if $r^2\le m$. In particular, all of these spheres are not $W_\xi$-stable.

Proof

For the standard sphere $S^m(r)\subset {{\mathbb {R}}}^{m+1}\subset {{\mathbb {R}}}^{m+p}$, we have $x^\top =0$, $h=-\frac{1}{r^2}gx$ and hence $\xi =\left( -\frac{m}{r^2}+1\right) x$. It follows that $x-\xi =\frac{m}{r^2}x$ and

$$\begin{aligned} L=\triangle ^\bot _{S^m(r)}+\left\langle h_{ij},\cdot \right\rangle h_{ij}+1=\triangle ^\bot _{S^m(r)}+\frac{m}{r^4}\left\langle x,\cdot \right\rangle x+1,\quad \tilde{{\mathcal {L}}}=\triangle _{S^m(r)}. \end{aligned}$$

In particular, $L(x)=\frac{1}{r^2}(m+r^2)x$ and, for any parallel normal vector field N orthogonal to x, $L(N)=N$. Let $e_{m+2},\ldots ,e_{m+p}$ be an orthonormal constant basis of the subspace $({\mathrm{Span\,}}\{TS^m(r),x\})^\bot \subset {{\mathbb {R}}}^{m+p}$. Then, $e_{m+1}:\equiv \frac{1}{r} x,e_{m+2},\ldots ,e_{m+p}$ is an orthonormal normal frame field of $S^m(r)$ and

$$\begin{aligned} L(e_{m+1})=\frac{1}{r^2}(m+r^2)e_{m+1},\quad L(e_\alpha )=e_\alpha ,\quad \alpha =m+2,\ldots ,m+p. \end{aligned}$$

(6.12)

Now for any $\eta \in \Gamma (T^\bot S^m(r))$ we can write

$$\begin{aligned} \eta =\sum _{\alpha }\eta ^\alpha e_\alpha \text { with } \eta ^\alpha \in C^\infty (S^m(r)), \ m+1\le \alpha \le m+p. \end{aligned}$$

Then by (5.4) and (6.12)

$$\begin{aligned} L(\eta )&=\sum _\alpha (\tilde{{\mathcal {L}}}(\eta ^\alpha )) e_\alpha +\eta ^\alpha L(e_\alpha ) \\&=(\triangle _{S^m(r)}\eta ^{m+1}) e_{m+1}+\eta ^{m+1} L(e_{m+1})+\sum _{\alpha \ge m+2}((\triangle _{S^m(r)}\eta ^\alpha ) e_\alpha +\eta ^\alpha L(e_\alpha ))\\&=\left( \tilde{L}+\frac{m}{r^2}\right) \eta ^{m+1} e_{m+1} +\sum _{\alpha \ge m+2}\tilde{L}(\eta ^\alpha ) e_\alpha \end{aligned}$$

where $\tilde{L}=\triangle _{S^m(r)}+1$. Furthermore, let $\lambda _k$, $k\ge 0$ be the eigenvalues of $\tilde{L}$ and write $\eta ^\alpha =\sum _{k\ge 0}\eta ^\alpha _k$ for some eigenfunctions $\eta ^\alpha _k$ satisfying $\tilde{L}(\eta ^\alpha _k)=-\lambda _k\eta ^\alpha _k$, $k\ge 0$.

It is well known that the eigenvalues of $-\triangle _{S^m(r)}$ is $\frac{k(m+k-1)}{r^2}$, $k\ge 0$, so that

$$\begin{aligned} \lambda _k=\frac{k(m+k-1)}{r^2}-1, \text { for }k=0,1,2,\ldots , \end{aligned}$$

with constants being the eigenfunctions corresponding to $k=0$. But by (6.1), $\int _{S^m(r)}\eta ^\alpha e^{-f}\mathrm{d}V_{S^m(r)}=0$ which implies that $\eta ^\alpha _0=0$. Therefore,

$$\begin{aligned} Q(\eta ,\eta )&=-\int _{S^m(r)}\left\langle \eta , L(\eta )\right\rangle e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&=-\int _{S^m(r)}\eta ^{m+1}\left( \tilde{L}+\frac{m}{r^2}\right) \eta ^{m+1} e^{-f}\mathrm{d}V_{S^m(r)}+\sum _{\alpha \ge m+2}\int _{S^m(r)}\eta ^\alpha (-\tilde{L}\eta ^\alpha ) e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&=\sum _{k\ge 1}\int _{S^m(r)}\left( \frac{k(m+k-1)}{r^2}-\frac{1}{r^2}(m+r^2)\right) |\eta ^{m+1}_k|^2 e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&\quad +\sum _{\alpha \ge m+2,k\ge 1}\int _{S^m(r)}\left( \frac{k(m+k-1)}{r^2}-1\right) |\eta ^\alpha _k|^2 e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&=-\int _{S^m(r)}|\eta ^{m+1}_1|^2 e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&\quad +\sum _{k\ge 2}\int _{S^m(r)}\left( \frac{k(m+k-1)}{r^2}-\frac{1}{r^2}(m+r^2)\right) |\eta ^{m+1}_k|^2 e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&\quad +\sum _{\alpha \ge m+2,k\ge 1}\int _{S^m(r)}\left( \frac{k(m+k-1)}{r^2}-1\right) |\eta ^\alpha _k|^2 e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&\ge -\int _{S^m(r)}|\eta ^{m+1}_1|^2 e^{-f}\mathrm{d}V_{S^m(r)}+\left( \frac{m+2}{r^2}-1\right) \sum _{k\ge 2}\int _{S^m(r)}|\eta ^{m+1}_k|^2 e^{-f}\mathrm{d}V_{S^m(r)}\nonumber \\&\quad +\left( \frac{m}{r^2}-1\right) \sum _{\alpha \ge m+2,k\ge 1}\int _{S^m(r)}|\eta ^\alpha _k|^2 e^{-f}\mathrm{d}V_{S^m(r)}. \end{aligned}$$

(6.13)

Define

$$\begin{aligned} V_{\lambda _1}=\{\varphi \in C^\infty (S^m(r));\ \triangle _{S^m(r)}\varphi =-\frac{m}{r^2}\varphi \},\quad \tilde{V}_{\lambda _1}=\{\varphi e_{m+1};\ \varphi \in V_{\lambda _1}\}. \end{aligned}$$

Then $\dim \tilde{V}_{\lambda _1}=\dim V_{\lambda _1}$ and the left side is well known to be $m+1$. It is not hard to see from (6.13) that Q is negative definite on $\tilde{V}_{\lambda _1}$, and thus, ${\mathrm{ind}}(S^m(r))\ge m+1$ with the equality holding if and only if $\frac{m}{r^2}-1\ge 0$, that is, $r^2\le m$. $\square$

7 The uniqueness problem for complete W-stable $\xi$-submanifolds

It is interesting to know whether or not m-planes are the only $W_\xi$-stable $\xi$-submanifolds. We shall start to deal with this problem in this section. To make things more clear, we would better use the standard weighted volume functional $V_w$ for immersed submanifolds, which is a special case of either $V_\xi$ or ${{\bar{V}}}_\xi$ with $\xi \equiv 0$:

$$\begin{aligned} V_w(x)\equiv V_0(x)=\int _Me^{-\frac{1}{2}|x|^2}\mathrm{d}V_x,\quad x\in {{\mathcal {M}}}. \end{aligned}$$

Then, the same argument as in the proof of Theorem 4.1, Theorem 4.3 and Corollary 4.4 easily lead to the following

Proposition 7.1

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a $\xi$-submanifold. Then for any VP-variation of x, we have

$$\begin{aligned} V_w'(t)&=-\int _M \left\langle H_{t}+x_{t}^\bot ,\frac{\partial F}{\partial t}\right\rangle e^{-\frac{1}{2}|x_t|^2}\mathrm{d}V_t,\quad x_t\in {{\mathcal {M}}}, \end{aligned}$$

(7.1)

$$\begin{aligned} V_w''(0)&=-\int _M\Big (\left\langle \triangle _{M}^\bot (\eta )-D^\bot _{x^\top }\eta + g^{ik}g^{jl}\left\langle h_{ij},\eta \right\rangle h_{kl}+\eta ,\eta \right\rangle \Big )e^{-\frac{1}{2}|x|^2}\mathrm{d}V. \end{aligned}$$

(7.2)

By making applications of (7.1) and (7.2), we can generalize the conventional extreme value problem with conditions to our situation. For example, we have by Definition 6.1 and (7.1):

Corollary 7.2

(see [27] for the hypersurface case). An immersion $x\in {{\mathcal {M}}}$ is a $\xi$-submanifold if and only if it is a critical point of $V_w$ under the VP-variations (the “critical point with condition”).

Now we introduce the concept of W-stability for $\xi$-submanifolds, which can be viewed as the “conditional” critical points of $V_w$.

Definition 7.1

A $\xi$-submanifold $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is called W-stable if it has a finite standard weighted volume $V_w(x)$ and $V_w''(0)\ge 0$ for all VP-variations of x.

In other words, the W-stability is exactly the $W_0$-stability, a typical one to the $W_\xi$-stability: just put $\xi =0$ in the functional $V_\xi$. In this sense our main theorem can be stated as follows:

Theorem 7.3

Let $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ be a complete and properly immersed $\xi$-submanifold with flat normal bundle. Then, x is W-stable if and only if $x(M^m)$ is an m-plane.

To prove this theorem, we shall extend the main idea in [27], originally applied for the hypersurface case, to our higher codimension case here by solving some certain technical problems. Clearly, we only need to prove the necessity part of Theorem 7.3. For this, we can first make use of the universal covering if necessary to assume that $M^m$ is simply connected. Then that x has a flat normal bundle implies the existence of a parallel orthonormal normal frame $\{e_\alpha ;\ m+1\le \alpha \le m+p\}$. Furthermore, from (5.1) or (7.2) we have

$$\begin{aligned} {{\mathcal {L}}}=\triangle ^\bot _{M^m}-D^\bot _{x^\top },\quad L={{\mathcal {L}}}+\left\langle h_{ij},\cdot \right\rangle h_{ij} +1,\quad \tilde{{\mathcal {L}}}=\triangle _{M^m} -\nabla _{x^\top }. \end{aligned}$$

(7.3)

Lemma 7.4

Let x be a $\xi$-submanifold. Then for any constant vector $v\in {{\mathbb {R}}}^{m+p}$ and any parallel normal vector field N, we have

$$\begin{aligned} \tilde{{\mathcal {L}}}\left\langle v,N\right\rangle =-\left\langle A_N,A_{v^\bot }\right\rangle \equiv -\left\langle h_{ij},N\right\rangle \left\langle h_{ij},v\right\rangle ,\quad L(v^\bot )= v^\bot \end{aligned}$$

(7.4)

where $v^\top$ and $v^\bot$ are the orthogonal projections of the vector v on $TM^m$ and $T^\bot M^m$, respectively.

Proof

By using Weingarten formula and the equality that $D^\bot (H+x^\bot )\equiv 0$, we find

$$\begin{aligned} \tilde{{\mathcal {L}}}\left\langle v,N\right\rangle&=\triangle _{M^m}\left\langle v,N\right\rangle -\nabla _{x^\top }\left\langle v,N\right\rangle \\&=(\left\langle v,-A_N(e_i)\right\rangle )_{,i}-\left\langle v,-A_N(x^\top )\right\rangle \\&=-\left\langle h_{iji},N\right\rangle \left\langle v,e_j\right\rangle -\left\langle h_{ij},N\right\rangle \left\langle v,e_j\right\rangle _{,i}+\left\langle v,A_N(x^\top )\right\rangle \\&=\left\langle x,N\right\rangle _j\left\langle v,e_j\right\rangle -\left\langle h_{ij},N\right\rangle \left\langle v,h_{ji}\right\rangle +\left\langle v,A_N(x^\top )\right\rangle \\&=-\left\langle x,A_N(e_j)\right\rangle \left\langle v,e_j\right\rangle -\left\langle A_N,A_{v^\bot }\right\rangle +\left\langle v,A_N(x^\top )\right\rangle \\&=-\left\langle x^\top ,A_N(v^\top )\right\rangle -\left\langle A_N,A_{v^\bot }\right\rangle +\left\langle A_N(v^\top ),x^\top \right\rangle \\&=-\left\langle A_N,A_{v^\bot }\right\rangle . \end{aligned}$$

The second equality follows directly from (5.3), (5.4) and the first equality in (7.4).$\square$

Lemma 7.5

For any $\eta =e_\alpha +v^\bot$, $v\in {{\mathbb {R}}}^{m+p}$, it holds that

$$\begin{aligned} Q(\phi \eta ,\phi \eta )\le -\int _M \phi ^2|\eta |^2e^{-f}\mathrm{d}V+\int _M |\nabla \phi |^2(|\eta |^2+|v^\top |^2)e^{-f}\mathrm{d}V,\quad \forall \,\phi \in C^\infty _0(M^m),\nonumber \\ \end{aligned}$$

(7.5)

where and hereafter we denote $f=\frac{1}{2}|x|^2$.

Proof

By (5.3) and (7.4),

$$\begin{aligned} L(\eta )=L(e_\alpha +v^\bot )=e_\alpha +h^\alpha _{ij}h_{ij}+v^\bot =\eta +h^\alpha _{ij}h_{ij}. \end{aligned}$$

It follows from (5.9) that

$$\begin{aligned}&Q(\phi \eta ,\phi \eta )=-\int _M \left\langle \phi \eta ,L(\phi \eta )\right\rangle e^{-f}\mathrm{d}V\nonumber \\&\quad =-\int _M\phi ^2\left\langle \eta ,L(\eta )\right\rangle e^{-f}\mathrm{d}V+\int _M |\nabla \phi |^2|\eta |^2 e^{-f}\mathrm{d}V\nonumber \\&\quad =-\int _M \phi ^2\left\langle \eta ,\eta +h^\alpha _{ij}h_{ij}\right\rangle e^{-f}\mathrm{d}V+\int _M |\nabla \phi |^2|\eta |^2 e^{-f}\mathrm{d}V\nonumber \\&\quad =-\int _M \phi ^2|\eta |^2 e^{-f}\mathrm{d}V -\int _M \phi ^2 h^\alpha _{ij}\left\langle h_{ij},e_\alpha +v^\bot \right\rangle e^{-f}\mathrm{d}V+\int _M |\nabla \phi |^2|\eta |^2 e^{-f}\mathrm{d}V. \end{aligned}$$

(7.6)

On the other hand, by (5.1) and (5.5)

$$\begin{aligned}&\int _M\phi ^2\left\langle e_\alpha ,v^\bot \right\rangle e^{-f}\mathrm{d}V=\int _M\phi ^2\left\langle e_\alpha , L(v^\bot )\right\rangle e^{-f}\mathrm{d}V\\&\quad =\int _M \phi ^2\left\langle e_\alpha ,\left\langle h_{ij},v^\bot \right\rangle h_{ij}+v^\bot \right\rangle e^{-f}\mathrm{d}V +\int _M\left\langle \phi ^2 e_\alpha ,{{\mathcal {L}}}v^\bot \right\rangle e^{-f}\mathrm{d}V\\&\quad =\int _M\phi ^2\left\langle e_\alpha ,v^\bot \right\rangle e^{-f}\mathrm{d}V+\int _M\phi ^2h^\alpha _{ij}\left\langle h_{ij}, v^\bot \right\rangle e^{-f}\mathrm{d}V-\int _M \left\langle D^\bot (\phi ^2e_\alpha ),D^\bot v^\bot \right\rangle e^{-f}\mathrm{d}V\\&\quad =\int _M\phi ^2\left\langle e_\alpha ,v^\bot \right\rangle e^{-f}\mathrm{d}V+\int _M\phi ^2h^\alpha _{ij}\left\langle h_{ij}, v^\bot \right\rangle e^{-f}\mathrm{d}V-2\int _M \phi \left\langle (\nabla \phi )e_\alpha ,-d(v^\top )\right\rangle e^{-f}\mathrm{d}V\\&\quad =\int _M\phi ^2\left\langle e_\alpha ,v^\bot \right\rangle e^{-f}\mathrm{d}V+\int _M\phi ^2h^\alpha _{ij}\left\langle h_{ij}, v^\bot \right\rangle e^{-f}\mathrm{d}V+2\int _M \phi h^\alpha (\nabla \phi ,v^\top )e^{-f}\mathrm{d}V, \end{aligned}$$

implying that

$$\begin{aligned}&\left| \int _M\phi ^2h^\alpha _{ij}\left\langle h_{ij}, v^\bot \right\rangle e^{-f}\mathrm{d}V\right| =\left| 2\int _M \phi h^\alpha (\nabla \phi ,v^\top )e^{-f}\mathrm{d}V\right| \\&\quad \le 2\int _M |\phi ||h^\alpha ||\nabla \phi ||v^\top |e^{-f}\mathrm{d}V\le \int _M \phi ^2|h^\alpha |^2e^{-f}\mathrm{d}V+\int _M|\nabla \phi |^2 |v^\top |^2e^{-f}\mathrm{d}V. \end{aligned}$$

Inserting this into (7.6) we complete the proof. $\square$

Define

$$\begin{aligned} W={\mathrm{Span\,}}_{{\mathbb {R}}}\{e_\alpha \},\quad V^\top =\{v^\top ; v\in {{\mathbb {R}}}^{m+p}\},\quad V^\bot =\{v^\bot ; v\in {{\mathbb {R}}}^{m+p}\}. \end{aligned}$$

(7.7)

Then W is the space of parallel normal fields of x and $p\le \dim V^\bot \le m+p$.

Lemma 7.6

Denote

$$\begin{aligned} V^\bot _0=\{v^\bot ={\mathrm{const}};\ v\in {{\mathbb {R}}}^{m+p}\}. \end{aligned}$$

(7.8)

Then $W\cap V^\bot =V^\bot _0$.

Proof

For any $\eta \in W\cap V^\bot$, we have $\eta =v^\bot =c^\alpha e_\alpha$ for some $v\in {{\mathbb {R}}}^{m+p}$ and $c^\alpha \in {{\mathbb {R}}}$. Then it follows from (5.3) and (5.4) that

$$\begin{aligned} v^\bot =L(v^\bot )=c^\alpha L(e_\alpha )=c^\alpha (e_\alpha +h^\alpha _{ij}h_{ij})=v^\bot +c^\alpha h^\alpha _{ij}h_{ij} \end{aligned}$$

implying that $c^\alpha h^\alpha _{ij}h_{ij}=0$. Multiplying this with $v^\bot =c^\alpha e_\alpha$ it follows that

$$\begin{aligned} \left\langle h,v^\bot \right\rangle ^2=\sum _{i,j,\alpha ,\beta }c^\alpha c^\beta h^\alpha _{ij}h^\beta _{ij}=0. \end{aligned}$$

Thus $\left\langle h,v^\bot \right\rangle =0$ or equivalently $A_{v^\bot }=0$ which with the fact that $v^\bot$ is parallel in the normal bundle shows that $v^\bot$ must be a constant vector.

The inverse part is trivial. $\square$

Define

$$\begin{aligned} \Gamma ^{\infty ,2}_w(T^\bot M^m):=\{\eta \in \Gamma (T^\bot M);\ \int _M|\eta |^2 e^{-f}\mathrm{d}V<+\infty \}, \end{aligned}$$

on which there is a standard $L^2_w$ inner product $(\cdot ,\cdot )$ by

$$\begin{aligned} (\eta _1,\eta _2):=\int _M\left\langle \eta _1,\eta _2\right\rangle e^{-f}\mathrm{d}V,\quad \forall \,\eta _1,\eta _2\in \Gamma ^{\infty ,2}_w(T^\bot M^m), \end{aligned}$$

giving the corresponding $L^2_w$-norm $\Vert \cdot \Vert _{2,w}$. The $L^2_w$ inner product $(\cdot ,\cdot )$ and $L^2_w$-norm $\Vert \cdot \Vert _{2,w}$ for all weighted square integrable tangent vector fields and functions on $M^m$ are defined in the same way. In particular, for a constant c, we have $\Vert c\Vert ^2_{2,w}=c^2\int _Me^{-f}\mathrm{d}V$.

Let $V^\bot _1$ be the orthogonal complement of $V^\bot _0$ in $V^\bot$ with respect to the $L^2_w$ inner product, and define $\tilde{V}=W\oplus V^\bot _1$ as subspaces of $\Gamma ^{\infty ,2}_w(T^\bot M^m)$. So for any $\eta \in \tilde{V}$ we can write $\eta =w+v^\bot$ for a unique $w\in W$ and some $v\in {{\mathbb {R}}}^{m+p}$ such that $\eta =w+v^\bot$ where v may not be unique. Since $\dim W=p$ and $\dim V^\bot _1\le \dim V^\bot \le m+p$, we have $\dim \tilde{V}<+\infty$. Fix a basis $\{w_a+v^\bot _a;\ 1\le a\le \dim \tilde{V}\}$ for $\tilde{V}$ such that $\Vert w_a\Vert ^2_{2,w}+\Vert v_a\Vert ^2_{2,w}=1$ for $1\le a\le \dim \tilde{V}$. Define

$$\begin{aligned} {{\mathbb {S}}}=\{\eta =\sum _a\eta ^a(w_a+v^\bot _a);\ \sum _a (\eta ^a)^2=1\}\subset \tilde{V}. \end{aligned}$$

Then the finiteness of $\dim \tilde{V}$ implies that ${{\mathbb {S}}}$ is compact. Note that for any $\eta \in {{\mathbb {S}}}$, $\eta$ can not be zero.

Now we consider the compact case and prove the following

Proposition 7.7

Any compact $\xi$-submanifold with parallel normal bundle can not be W-stable.

Proof

It suffices to show that both of the following two are true:

• (1) Q is negative definite on $\tilde{V}$ and, consequently, is negative definite on $V^\bot _1$;

• (2) $\dim V^\bot _1>0$.

In fact, the conclusion (1) follows directly from Lemma 7.5 by choosing $\phi \equiv 1$; while conclusion (2) follows from the fact that the converse of (2) would imply that $M^m={{\mathbb {R}}}^m$, by the argument at the end of this paper, which contradicts the compactness assumption. $\square$

Next we consider the non-compact case and thus assume that $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ is a complete and non-compact $\xi$-submanifold.

Let o be a fixed point of M and ${{\bar{o}}}=x(o)$. For any $R>0$, we define ${{\bar{B}}}_R({{\bar{o}}})=\{x\in {{\mathbb {R}}}^{m+p};\ |x-{{\bar{o}}}|\le R\}$ and introduce a cut-off function ${{\bar{\phi }}}_R$ as follows (cf. [27]):

$$\begin{aligned} {{\bar{\phi }}}_R(x)= {\left\{ \begin{array}{ll} 1,&{} x\in {{\bar{B}}}_R({{\bar{o}}});\\ 1-\frac{1}{R}(|x-{{\bar{o}}}|-R),&{} x\in {{\bar{B}}}_{2R}({{\bar{o}}})\backslash {{\bar{B}}}_R({{\bar{o}}});\\ 0,&{}x\in {{\mathbb {R}}}^{m+p}\backslash {{\bar{B}}}_{2R}({{\bar{o}}}). \end{array}\right. } \end{aligned}$$

(7.9)

For the given immersion $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$, let $\phi _R={{\bar{\phi }}}_R\circ x\in C^\infty (M^m)$ and $B_R(o)=x^{-1}({{\bar{B}}}_R({{\bar{o}}}))$. Then, $B_R(o)$ is compact since x is properly immersed. In particular, $\phi _R$ is compactly supported. Furthermore, it is easily seen that $|\nabla \phi _R|\le |D{{\bar{\phi }}}_R|\le \frac{1}{R}$.

Lemma 7.8

There is a large $R_0>0$ such that

$$\begin{aligned} \int _{B_R(o)}|\eta |^2e^{-f}\mathrm{d}V\ge \int _{B_{R_0}(o)}|\eta |^2e^{-f}\mathrm{d}V>0,\quad \forall \,\eta \in {{\mathbb {S}}},\quad \forall \,R\ge R_0. \end{aligned}$$

Proof

If the lemma is not true, then one can find a sequence $\{\eta _j\}\subset {{\mathbb {S}}}$ such that

$$\begin{aligned} \int _{B_j(o)}|\eta _j|^2e^{-f}\mathrm{d}V=0, \quad j=1,2,\ldots . \end{aligned}$$

By the compactness of ${\mathbb {S}}$, there exists a subsequence $\{\eta _{j_k}\}$ which is convergent to some $\eta _0\in {{\mathbb {S}}}$. For any $R>0$, there exists some $K>0$ such that $j_k>R$ for all $k>K$. It follows that

$$\begin{aligned} \int _{B_R(o)}|\eta _0|^2e^{-f}\mathrm{d}V=\lim _{k\rightarrow +\infty }\int _{B_R(o)}|\eta _{j_k}|^2e^{-f}\mathrm{d}V =0 \end{aligned}$$

which implies that

$$\begin{aligned} \int _M|\eta _0|^2e^{-f}\mathrm{d}V=\lim _{R\rightarrow +\infty }\int _{B_R(o)}|\eta _0|^2e^{-f}\mathrm{d}V =0. \end{aligned}$$

Thus we have $\eta _0=0$ contradicting to the fact that $\eta _0\in {{\mathbb {S}}}$. $\square$

For each $R>0$, define

$$\begin{aligned} m_R:=\min _{\eta \in {{\mathbb {S}}}}\{\int _M\phi _R^2|\eta |^2e^{-f}\mathrm{d}V\},\quad M_R=\max _{\eta \in {{\mathbb {S}}}}\{\int _M\phi _R^2|\eta |^2e^{-f}\mathrm{d}V\}. \end{aligned}$$

(7.10)

Clearly,

$$\begin{aligned} M_R\le C:\equiv \max _{\eta \in {{\mathbb {S}}}}\int _M|\eta |^2e^{-f}\mathrm{d}V<+\infty . \end{aligned}$$

(7.11)

Moreover, $m_R$ is increasing with respect to R which together with Lemma 7.8 gives that

$$\begin{aligned} m_R\ge m_{R_0}>0,\quad \forall \,R\ge R_0. \end{aligned}$$

(7.12)

Lemma 7.9

There exists a large $R_0$, such that

$$\begin{aligned} \dim \phi _R\tilde{V} =\dim \tilde{V},\quad \dim \phi _RV^\bot _1=\dim V^\bot _1,\quad R\ge R_0; \end{aligned}$$

(7.13)

Furthermore, Q is negative definite on $\phi _R\tilde{V} \supset \phi _RV^\bot _1$.

Proof

First, we prove $\dim \phi _R\tilde{V}=\dim \tilde{V}$ for all $R\ge R_0$ if $R_0$ is large enough. For a given $R>0$, consider the surjective linear map

$$\begin{aligned} \Phi _R:\tilde{V}\rightarrow \phi _R \tilde{V},\quad \eta \mapsto \Phi _R(\eta ):=\phi _R\eta ,\quad \forall \, \eta \in \tilde{V}. \end{aligned}$$

We claim that, when $R_0$ is large enough, the kernel $\ker \Phi _{R_0}$ of $\Phi _{R_0}$ must be trivial. In fact, if it is not the case, there should be a nonzero sequence $\{\eta _j\in \tilde{V}\}$ such that $\phi _j\eta _j=0$. By writing $\eta _j=\sum _a\eta ^a_j(w_a+v^\bot _a)$, we can define $\tilde{\eta }_j=\frac{\eta _j}{\sqrt{\sum _a (\eta ^a_j)^2}}$. Then $\phi _j\tilde{\eta }_j=0$, and $\{\tilde{\eta }_j\}$ is contained in ${\mathbb {S}}$. Then the compactness of ${\mathbb {S}}$ assures that, by passing to a subsequence if possible, we can assume that $\tilde{\eta }_j\rightarrow \tilde{\eta }_0\in {{\mathbb {S}}}$. Consequently, we have $\tilde{\eta }_0=\lim _{j\rightarrow +\infty }\phi _j\tilde{\eta }_j=0$ which is not possible. So there must be a large $R_0>0$ such that $\ker \Phi _{R_0}=0$ and the claim is proved.

For any $R\ge R_0$, it is easily seen that $\ker \Phi _R\subset \ker \Phi _{R_0}$ which implies that $\ker \Phi _R=0$ and $\phi _R\tilde{V}\cong \tilde{V}$. In particular, $\dim \phi _R\tilde{V}=\dim \tilde{V}$.

That $\dim \phi _RV^\bot _1=\dim V^\bot _1$ follows in the same way.

Next we are to find a larger $R\ge R_0$ such that Q is negative definite on $\phi _R\tilde{V}$. For this, we first note that $|\nabla \phi _R|$ supports in $B_{2R}(o)\backslash B_R(o)$ and $|\nabla \phi _R|\le \frac{1}{R}$, and then use Lemma 7.5 to conclude that, for all $\eta \in {{\mathbb {S}}}$

$$\begin{aligned} Q(\phi _R\eta ,\phi _R\eta )&\le -\int _M\phi _R^2 |\eta |^2 e^{-f}\mathrm{d}V+\int _M|\nabla \phi _R|^2(|\eta |^2+|v^\top |^2)e^{-f}\mathrm{d}V\\&\le -\int _M\phi _R^2 |\eta |^2 e^{-f}\mathrm{d}V+\frac{1}{R^2}\int _{B_{2R}(0)\backslash B_R(0)}(|\eta |^2+|v^\top |^2)e^{-f}\mathrm{d}V\\&\le -\int _M\phi _R^2 |\eta |^2 e^{-f}\mathrm{d}V+\frac{3}{R^2}\dim \tilde{V}. \end{aligned}$$

Therefore, by (7.10)–(7.12) and Lemma 7.8, there must be an $R_0$ large enough such that $Q(\phi _R\eta ,\phi _R\eta )<0$ for all $\eta \in {{\mathbb {S}}}$, $R\ge R_0$. Then the conclusion that Q is negative definite on $\phi _R\tilde{V}$ follows directly from the bilinearity of Q. $\square$

Lemma 7.10

Under the complete and non-compact assumption, we have

$$\begin{aligned} V^\bot _1=0 \text { or equivalently, }V^\bot =V^\bot _0. \end{aligned}$$

(7.14)

Proof

Let $W^\bot$ be the orthogonal complement of W in the space $\Gamma ^{\infty ,2}_w(T^\bot M^m)$ of $L^2_w$-smooth normal sections. For any given $R>0$, define a subspace

$$\begin{aligned} W^\bot (\phi _R\tilde{V}):=W^\bot \cap (\phi _R\tilde{V}) \end{aligned}$$

of $W^\bot$ and a linear map $\Psi _R:\phi _RV^\bot _1\rightarrow W^\bot (\phi _R\tilde{V})$ by

$$\begin{aligned} \phi _Rv^\bot \mapsto \Psi _R(\phi _Rv^\bot ):=\phi _Rv^\bot -\frac{\int _M\left\langle \phi _R v^\bot ,e_\alpha \right\rangle e^{-f}\mathrm{d}V}{\int _M\phi _R e^{-f}\mathrm{d}V}\phi _R e_\alpha ,\quad \,\forall v^\bot \in V^\bot _1. \end{aligned}$$

Claim: There must be a large $R>0$ such that $\ker \Psi _R=0$.

In fact, if this is not true, then we can find a sequence $\{v^\bot _j\}\subset V^\bot _1$ with $\phi _jv^\bot _j\ne 0$ and $\Psi _j(\phi _jv^\bot _j)=0$ for each $j=1,2,\ldots$. It follows that $v^\bot _j\ne 0$, $j=1,2,\ldots$. Define

$$\begin{aligned} \tilde{v}^\bot _j:=\frac{v^\bot _j}{\Vert v^\bot _j\Vert _{2,w}},\quad j=1,2,\ldots . \end{aligned}$$

Then $\Psi _j(\phi _j\tilde{v}^\bot _j)=0$, $j=1,2,\ldots$. Without loss of generality, we can assume that $\tilde{v}^\bot _j\rightarrow \tilde{v}^\bot _0$. Then $\tilde{v}^\bot _0\in V^\bot _1$ and $\Vert \tilde{v}^\bot _0\Vert _{2,w}=1$.

On the other hand, from $\Psi _j(\phi _j\tilde{v}^\bot _j)=0$ ($j=1,2,\ldots )$ it follows that

$$\begin{aligned} \phi _j\tilde{v}^\bot _j=\frac{\int _M\left\langle \phi _j\tilde{v}^\bot _j,e_\alpha \right\rangle e^{-f}\mathrm{d}V}{\int _M\phi _j e^{-f}\mathrm{d}V}\phi _j e_\alpha ,\quad j=1,2,\ldots , \end{aligned}$$

implying that

$$\begin{aligned} \Vert \phi _j\tilde{v}^\bot _j\Vert ^2_{2,w}=\frac{\int _M\left\langle \phi _j\tilde{v}^\bot _j,e_\alpha \right\rangle e^{-f}\mathrm{d}V}{\int _M\phi _j e^{-f}\mathrm{d}V}(\phi _j e_\alpha ,\phi _j\tilde{v}^\bot _j),\quad j=1,2,\ldots . \end{aligned}$$

(7.15)

But it is clear that $\phi _j\tilde{v}^\bot _j\rightarrow \tilde{v}^\bot _0$ when $j\rightarrow +\infty$ since

$$\begin{aligned} \Vert \phi _j\tilde{v}^\bot _j-\tilde{v}^\bot _0\Vert _{2,w}&\le \Vert \phi _j(\tilde{v}^\bot _j-\tilde{v}^\bot _0)\Vert _{2,w} +\Vert (\phi _j-1) \tilde{v}^\bot _0 \Vert _{2,w}\\&\le \Vert \tilde{v}^\bot _j-\tilde{v}^\bot _0\Vert _{2,w} +\Vert \phi _j-1\Vert _{2,w}\rightarrow 0,\quad j\rightarrow +\infty . \end{aligned}$$

Let $j\rightarrow +\infty$ in (7.15) then we obtain

$$\begin{aligned} \Vert \tilde{v}^\bot _0\Vert ^2_{2,w}=\frac{\int _M\left\langle \tilde{v}^\bot _0,e_\alpha \right\rangle e^{-f}\mathrm{d}V}{\int _M e^{-f}\mathrm{d}V}(e_\alpha ,\tilde{v}^\bot _0)=0 \end{aligned}$$

because $\tilde{v}^\bot _0\in V^\bot _1$ is orthogonal to W, contradicting to the fact that $\Vert \tilde{v}^\bot _0\Vert _{2,w}=1$. So the claim is proved.

Thus by (7.13), when R is large enough it holds that

$$\begin{aligned} \dim V^\bot _1=\dim \phi _RV^\bot _1\le \dim W^\bot (\phi _R\tilde{V})\le {{\mathrm{ind}}}_W(Q) \end{aligned}$$

where ${{\mathrm{ind}}}_W(Q)$ denotes the W-stability index of Q. By the W-stability of x we have ${{\mathrm{ind}}}_W(Q)=0$, implying that $\dim V^\bot _1=0$ and thus $V^\bot _1=0$, which is equivalent to $V^\bot =V^\bot _0$. $\square$

Proof of Theorem 7.3

Using Proposition 7.7, we conclude that $x:M^m\rightarrow {{\mathbb {R}}}^{m+p}$ must be non-compact. Then by Lemma 7.10, we have a direct decomposition

$$\begin{aligned} {{\mathbb {R}}}^{m+p}=V^\top \oplus V^\bot \end{aligned}$$

where $V^\top$ now consists of all constant vectors in ${{\mathbb {R}}}^{m+p}$ that are tangent to $x_*TM^m$ at each point of $M^m$, while $V^\bot$ consists of all constant vectors in ${{\mathbb {R}}}^{m+p}$ that are normal to $x_*TM^m$ at each point of $M^m$. It then follows that $\dim V^\top \le m$ and $\dim V^\bot \le p$. Consequently,

$$\begin{aligned} m+p=\dim {{\mathbb {R}}}^{m+p}=\dim V^\top +\dim V^\bot \le m+p \end{aligned}$$

which implies that $\dim V^\top =m$ and $\dim V^\bot =p$. This is true only if $x(M^m)\equiv P^m$.

Theorem 7.3 is proved. $\square$