Abstract
We study selfexpanding solutions \(M^{m}\subset \mathbb {R}^{n}\) of the mean curvature flow. One of our main results is, that complete mean convex selfexpanding hypersurfaces are products of selfexpanding curves and flat subspaces, if and only if the function A^{2}/H^{2} attains a local maximum, where A denotes the second fundamental form and H the mean curvature vector of M. If the principal normal ξ = H/H is parallel in the normal bundle, then a similar result holds in higher codimension for the function A^{ξ}^{2}/H^{2}, where A^{ξ} is the second fundamental form with respect to ξ. As a corollary we obtain that complete mean convex selfexpanders attain strictly positive scalar curvature, if they are smoothly asymptotic to cones of nonnegative scalar curvature. In particular, in dimension 2 any mean convex selfexpander that is asymptotic to a cone must be strictly convex.
1 Introduction
A smooth immersion \(F:M\to \mathbb {R}^{n}\) of a manifold M of dimension m into euclidean space is called a selfexpander of the mean curvature flow, if it satisfies the equation
where H is the mean curvature vector of the immersion, λ is a positive constant and where ^{⊥} denotes the orthogonal projection onto the normal bundle of M.
Selfexpanders arise naturally when one considers solutions of graphical mean curvature flow. In the case of codimension 1 and under certain assumptions on the initial hypersurface at infinity, Ecker and Huisken [7] showed that the solutions of mean curvature flow of entire graphs in euclidean space exist for all times t > 0 and become asymptotically selfexpanding as \(t\to \infty \). Later Stavrou [32] proved such a result under the weaker assumption that the initial hypersurface attains a unique tangent cone at infinity. Rasul [25] showed that under an alternative condition at infinity and bounded gradient, the rescaled graphs converge to selfsimilar solutions but at a slower speed. Clutterbuck and Schnürer [5] considered graphical solutions to mean curvature flow and obtained a stability result for homothetically expanding solutions coming out of cones of positive mean curvature. It is expected that similar results hold for the mean curvature flow in higher codimension of entire graphs generated by contractions and area decreasing maps as studied in [26,27,28].
As was pointed out in [7] and [32], selfexpanders also arise as solutions of the mean curvature flow, if the initial submanifold is a cone. Moreover, in some situations uniqueness of selfexpanders is important for the construction of mean curvature flows starting from certain singular configurations [1]. Fong and McGrath [8] proved a Liouvilletype theorem for complete, meanconvex selfexpanders whose ends have decaying principal curvatures. Ding [6] studied selfexpanding solutions and their relationship to minimal cones. The space of asymptotically conical selfexpanders was studied in several papers by Bernstein and Wang, as for example in [2] and [3].
Cheng and Zhou [4] proved results for selfexpanders in higher codimension related to the spectrum of the drifted Laplacian. In higher codimension selfexpanders have been studied in particular for the Lagrangian mean curvature flow. In [20, 21, 24] new examples of Lagrangian selfexpanders were given. Lotay and Neves [22] proved that zeroMaslov class Lagrangian selfexpanders in \(\mathbb {C}^{n}\) that are asymptotic to a pair of planes intersecting transversely are locally unique for n > 2 and unique if n = 2. Further uniqueness results for Lagrangian selfexpanders asymptotic to the union of two transverse Lagrangian planes were shown by Imagi, Joyce and dos Santos [14].
Many of the above mentioned results show that the geometry of a selfexpander is strongly determined by its asymptotic structure at infinity. This is confirmed also by the main results of this paper. We show that the pinching quantityA^{H}^{2}/H^{4}, where A^{H} denotes the second fundamental form with respect to the mean curvature vector H is controlled by its geometry at infinity. In particular, this implies a number of uniqueness results for selfexpanders with a certain asymptotic behavior. Since selfexpanders are also minimal submanifolds with respect to a conformally flat Riemannian metric on \(\mathbb {R}^{n}\), the analysis of selfexpanders is very similar to that of classical minimal submanifolds. Therefore one expects also Bernstein type theorems for selfexpanders similar to the classical Bernstein theorems in higher codimension, for example as derived in [17,18,19, 29, 31].
2 Structure Equations for General Euclidean Submanifolds
Let \( F:M^{m}\to \mathbb {R}^{n}\) be a smooth immersion. We denote its pullback bundle by \(F^{\ast } T\mathbb {R}^{n}\) and the normal bundle by T^{⊥}M. The induced metric or first fundamental form g = F^{∗}〈⋅,⋅〉 on TM is given by
where \(dF\in {\Gamma }(F^{\ast }T\mathbb {R}^{n}\otimes T^{\ast } M)\) denotes the differential of F. The second fundmental form \(A\in {\Gamma }(F^{\ast }T\mathbb {R}^{n}\otimes T^{\ast } M\otimes T^{\ast } M)\) is defined by
Here and in the following all canonically induced full LeviCivita connections on product bundles over M will be denoted by ∇. We will sometimes use the connection on the normal bundle and on bundles formed from products with the normal bundle. These connections will be denoted by ∇^{⊥}.
By definition, A is a section in the bundle \(F^{\ast }T\mathbb {R}^{n}\otimes T^{\ast } M\otimes T^{\ast } M\) but it is well known that A is normal, i.e.,
This implies that
The mean curvature vector field H ∈ Γ(T^{⊥}M) is the trace of the second fundamental tensor. At p ∈ M we have
where \((e_{k})_{k=1,\dots ,m}\) denotes an arbitrary orthonormal basis of T_{p}M (it should be noted that many authors prefer to define H as \(\frac {1}{m}\text {trace}_{g}A\), for example this is done in [16]).
For any normal vector ξ ∈ T^{⊥}M we define the second fundamental form A^{ξ} and the scalar mean curvature H^{ξ} with respect to ξ by
The Riemannian curvature tensor on the tangent bundle will be denoted by R, whereas the curvature tensor of the normal bundle, considered as a 2form with values in End(T^{⊥}M) will be written as R^{⊥}.
We summarize the equations of Gauss, Ricci, Codazzi and Simons in the following proposition.
Proposition 1
Let M be an mdimensional smooth manifold and \(F:M\to \mathbb {R}^{n}\) be a smooth immersion. Then for any p ∈ M and any \(\xi \in {T}_{p}^{\perp } M\), v, w, u, z ∈ T_{p}M we have

(a)
Gauss:
$$ R(v,w,u,z)=\langle A(v,u),A(w,z)\rangle\langle A(v,z),A(w,u)\rangle. $$ 
(b)
Ricci:
$$ R^{\perp}(v,w)\xi=\sum\limits_{k=1}^{m} \left( A^{\xi}(w,e_{k}) A(v,e_{k})A^{\xi}(v,e_{k})A(w,e_{k})\right), $$where \((e_{k})_{k=1,\dots ,m}\) is an orthonormal basis of T_{p}M.

(c)
Codazzi:
$$ {\nabla}_{v}^{\perp}A(w,z)={\nabla}_{w}^{\perp}A(v,z). $$ 
(d)
Simons:
$$ \begin{array}{@{}rcl@{}} {\Delta}^{\perp} A(v,w) &=& (\nabla^{\perp})^{2}_{v,w}H+\sum\limits_{k=1}^{m}A^{H}(v,e_{k})A(w,e_{k})\\ &&+2\sum\limits_{k,l=1}^{m}\langle A(v,e_{k}),A(w,e_{l})\rangle A(e_{k},e_{l})\sum\limits_{k,l=1}^{m}\langle A(v,w),A(e_{k},e_{l})\rangle A(e_{k},e_{l})\\ &&\sum\limits_{k,l=1}^{m}\langle A(v,e_{k}),A(e_{k},e_{l})\rangle A(w,e_{l})\sum\limits_{k,l=1}^{m}\langle A(w,e_{k}),A(e_{k},e_{l})\rangle A(v,e_{l}). \end{array} $$
We set
Decomposing F = F^{⊥} + F^{⊤} into its normal and tangent components, the following equations are well known (see for example [30]):
and
3 Geometric Equations for Selfexpanders
Remark 1
We give some remarks.

(a)
A selfexpander that is minimal must be a cone. Hence, if the selfexpander is everywhere smooth and complete, then the only minimal selfexpanders are given by linear subspaces \(U\subset \mathbb {R}^{n}\).

(a)
Since the Riemannian product of two selfexpanders with the same constant λ is again a selfexpander, one observes that selfexpanders are not necessarily asymptotic to cones. For example the product of an expanding curve Γ with a linear subspace is such a special case. These selfexpanders will become important further below.
Let us now assume that the immersion is a selfexpander, i.e.,
for some positive^{Footnote 1} constant λ. From (1) and (2) we derive the following equations:
and
Taking a trace gives
If we take a scalar product with 2H we obtain
In addition, combining Simons’ identity with (3) we get
4 SelfExpanding Curves
In the case of curves \({\Gamma }\subset \mathbb {R}^{n}\) the equation for selfexpanders becomes a 2nd order system of ODEs. The existence and uniqueness theorem of Picard–Lindelöf implies that for any \(p,w\in \mathbb {R}^{n}\), w ≠ 0, there exists a maximal open interval I containing 0 and a uniquely determined curve \({\Gamma }:I\to \mathbb {R}^{n}\), parameterized proportional to arclength, such that
where \(\overrightarrow k\) denotes the curvature vector of Γ. If p, w are collinear, then clearly the straight lines passing through the origin in direction of w are the solutions. On the other hand, if p, w are linearly independent, then again by the uniqueness part in the theorem of Picard–Lindelöf the solution Γ must be a planar curve in the plane spanned by p, w. Since rotations around the origin map selfexpanders to selfexpanders, one may therefore without loss of generality consider only selfexpanding curves in \(\mathbb {R}^{2}\). For these curves the equation for selfexpanders becomes
where ξ is the unit normal along Γ obtained by rotating \({\Gamma }^{\prime }/\vert {\Gamma }^{\prime }\vert \) to the left by π/2 and k denotes the curvature function of Γ defined by \(\overrightarrow k=k\xi \). Solutions of (6) have been studied in great detail and are completely classified. Ishimura [15] showed that selfexpanding curves are asymptotic to a cone with vertex at the origin. It is easy to show (see [10, Theorem 3.20] or [12, Lemma 6.4]) that the function \(ke^{\frac {\lambda }{2}r^{2}}\) is constant along Γ, where r := Γ. The following description of selfexpanding curves can be found in [12]:
Proposition 2 (Halldorsson)
All selfexpanding curves \({\Gamma }\subset \mathbb {R}^{2}\) are convex, properly embedded and asymptotic to the boundary of a cone with vertex at the origin. They are graphs of even functions and form a onedimensional family parametrized by their distance r_{0} to the origin, which can take on any value in \([0,\infty )\).
Moreover, the total curvature \({\int \limits }_{\Gamma } k\) of the curves is given by π − α, where α denotes the opening angle of the asymptotic cone (Fig. 1).
In the sequel, any selfexpander \(M={\Gamma }\times \mathbb {R}^{m1}\subset \mathbb {R}^{m+1}\), where Γ is a nontrivial selfexpanding curve in \(\mathbb {R}^{2}\) (i.e., not a straight line), will be called a selfexpanding hyperplane (see Fig. 2). In particular selfexpanding hyperplanes are diffeomorphic to \(\mathbb {R}^{m}\).
5 Mean Convex SelfExpanding Hypersurfaces
In this section we will consider complete connected selfexpanding hypersurfaces. From (*) one observes that a smooth minimal selfexpander must be totally geodesic, hence a linear subspace. One of our main theorems is:
Theorem 1
Let \(M^{m}\subset \mathbb {R}^{m+1}\) be a smooth and complete connected selfexpander that is different from a linear subspace. Then the set {H≠ 0} is nonempty and the following statements are equivalent:

(a)
M is a selfexpanding hyperplane \({\Gamma }\times \mathbb {R}^{m1}\).

(b)
The function \(\frac {A^{2}}{H^{2}}\) attains a local maximum on the open set {H≠ 0}.
If one of these equivalent conditions is satisfied, then the set {H = 0} is empty and the function \(\frac {A^{2}}{H^{2}}\) is constant to 1.
Proof
Since M is a hypersurface we have A^{H}^{2} = H^{2}A^{2} and (4) becomes
Moreover, on the set where H is nonzero we have ∇^{⊥}H^{2} = ∇H^{2}. In addition, (5) simplifies to
from which we conclude
Finally, since
we derive in a similar way to the computations in [13] and [23]
If \(\frac {A^{2}}{H^{2}}\) attains a local maximum, the strong elliptic maximum principle implies that \(\frac {A^{2}}{H^{2}}\) is constant and that Q^{2} vanishes. Hence
Codazzi’s equation then implies that the tensor A ⊗∇^{⊥}H is fully symmetric, considered as a trilinear form an TM. We distinguish two cases.
Case 1
Suppose that ∇^{⊥}H = 0 everywhere, i.e., that H is constant. Since by assumption M is not a linear subspace this constant cannot be zero by Remark 1(a). Equation (7) then implies ∇^{⊥}A = 0. So all principal curvatures of M are constant and due to a well known theorem of Lawson, it follows that M is locally isometric to the product of a round sphere and a euclidean factor. Since those submanifolds are not selfexpanding this is impossible and Case 1 never occurs.
Case 2
Since Case 1 is impossible we may therefore assume that there exists a simply connected domain U ⊂ M where ∇^{⊥}H≠ 0. On this set we choose an orthonormal frame
Then from the symmetry of A ⊗∇^{⊥}H we obtain A(e_{j}, e_{k}) = 0, for any k ≥ 1 and j ≥ 2. Therefore, M has only one nonzero principal curvature on U and A^{2} = H^{2} on U. Let \({\mathcal{D}}:U\to TU\),
be the nullity distribution and \({\mathcal{D}}^{\perp }:=\text {span}\{e_{1}\}\) its orthogonal complement. Exactly as in [23] we have \(TU={\mathcal{D}}\oplus {\mathcal{D}}^{\perp }\) and conclude that both distributions \({\mathcal{D}}\), \({\mathcal{D}}^{\perp }\) are parallel so that by the de Rham decomposition theorem, U splits into the Riemannian product of a planar curve Γ and an (m − 1)dimensional euclidean factor. Since U is a selfexpander, the curve Γ must be part of a selfexpanding curve. It is well known that selfexpanding hypersurfaces are real analytic, therefore by completeness the local splitting implies the global splitting. This completes the proof. □
Recall that a hypersurface is called mean convex, if H > 0 everywhere. The next corollary shows that the pinching quantity A^{2}/H^{2} on mean convex selfexpanders is controlled by its asymptotic behavior at infinity.
Corollary 1
Let \(M\subset \mathbb {R}^{m+1}\) be a properly immersed mean convex selfexpanding hypersurface and suppose
where B(0, r) denotes the closed euclidean ball of radius r centered at the origin. Then one of the following cases holds:

(a)
\(\frac {A^{2}}{H^{2}}<\mu \) on all of M.

(b)
M is a selfexpanding hyperplane \({\Gamma }\times \mathbb {R}^{m1}\) and \(\frac {A^{2}}{H^{2}}\) is constant to μ = 1.
In particular, if μ≠ 1, then (a) holds.
Note, that μ = 1 does not exclude case (a).
Proof
Since M is mean convex, the function \(f:=\frac {A^{2}}{H^{2}}\) is well defined on all of M.
Step 1. We will first prove that f ≤ μ on M. Suppose there exists a point p ∈ M such that at p we have f(p) = μ + 𝜖 for some 𝜖 > 0. Choose r > 0 such that \(\sup _{M\setminus B(0,r)} f\le \mu +\epsilon /2\). Then p ∈ M ∩ B(0, r). Moreover, since M by assumption is properly immersed, the set K := M ∩ B(0, R) is compact. Hence the function f attains a local maximum on K. From Theorem 1 we conclude that f is constant to 1 and \(M={\Gamma }\times \mathbb {R}^{m1}\). Since f(p)≠μ this gives a contradiction.
Step 2. From Step 1 we know \(\sup _{M}f\le \mu \). Theorem 1 implies that either f < μ on all of M or f is constant to 1 and M is equal to a selfexpanding hyperplane. This completes the proof.
□
The last corollary can also be stated in the following form:
Corollary 2
For any properly immersed mean convex selfexpanding hypersurface \(M\subset \mathbb {R}^{m+1}\) we have
and equality occurs at some point p ∈ M, if and only if \(M={\Gamma }\times \mathbb {R}^{m1}\) is a selfexpanding hyperplane in which case \(\frac {A^{2}}{H^{2}}\) is constant to 1.
There exist some situations where the asymptotic behavior of \(\frac {A^{2}}{H^{2}}\) can be easily controlled, e.g. if the selfexpander is smoothly asymptotic to a cone.
Corollary 3
Any properly immersed mean convex selfexpanding surface \(M^{2}\subset \mathbb {R}^{3}\) that is smoothly asymptotic to a cone must be strictly convex.
Proof
On any 2dimensional cone the function \(\frac {A^{2}}{H^{2}}\) is constant to 1, therefore we conclude
The statement now follows from Gauß’ equation for the scalar curvature S = H^{2} −A^{2} and from Corollary 1 since the selfexpanding hyperplane is not smoothly asymptotic to a cone. □
There exists an extension of Corollary 3 to any dimension in the following sense.
Corollary 4
Any properly immersed mean convex selfexpanding hypersurface \(M\subset \mathbb {R}^{m+1}\) that is smoothly asymptotic to a cone with nonnegative scalar curvature must attain strictly positive scalar curvature.
Proof
Since the cone C has nonnegative scalar curvature S, we must have S = H^{2} −A^{2} ≥ 0 on C. On the other hand M is mean convex and asymptotic to C so that the cone must be mean convex as well. Therefore \(\frac {A^{2}}{H^{2}}\le 1\) on C and
Again from Corollary 1 and since the selfexpanding hyperplane is not smoothly asymptotic to a cone we conclude \(\frac {A^{2}}{H^{2}}< 1\) on M which by Gauß’ equation is equivalent to the statement that M has strictly positive scalar curvature. □
The second fundamental form of a hypersurface that is mean convex and of positive scalar curvature satisfies some nice properties. The proof of the next lemma is standard and follows from [9], see also Proposition 1(ii) in [11], where one has to choose σ_{1} := H and σ_{2} := S/2.
Lemma 1
Let \(M\subset \mathbb {R}^{m+1}\) be mean convex with positive scalar curvature S. Then the principal curvatures \(\lambda _{1},\dots ,\lambda _{m}\) of M satisfy λ_{i} < H for \(i=1,\dots ,m\).
A hypersurface \(M\subset \mathbb {R}^{m+1}\) is called kconvex, if at each point p ∈ M the sum of any k of the m principal curvatures \(\lambda _{1},\dots ,\lambda _{m}\) is positive. Obviously, mconvexity is the same as mean convexity and a strictly convex hypersurface is 1convex. Therefore Corollary 4 and Lemma 1 imply
Corollary 5
Any properly immersed mean convex selfexpanding hypersurface \(M\subset \mathbb {R}^{m+1}\) that is smoothly asymptotic to a cone with nonnegative scalar curvature must be (m − 1)convex.
Since a 3dimensional cone in \(\mathbb {R}^{4}\) has nonnegative scalar curvature, if it is convex, we conclude in particular
Corollary 6
Any properly immersed mean convex selfexpanding hypersurface \(M\subset \mathbb {R}^{4}\) that is smoothly asymptotic to a convex cone is 2convex.
There exist more results that can be obtained from Theorem 1 and its corollaries, for example
Corollary 7
Any properly immersed mean convex selfexpanding surface \(M\subset \mathbb {R}^{3}\) that is smoothly asymptotic to a selfexpanding hyperplane \({\Gamma }\times \mathbb {R}\) is a selfexpanding hyperplane.
Proof
Since M is smoothly asymptotic to \({\Gamma }\times \mathbb {R}\) we have
Corollary 1 implies that M is either equal to a selfexpanding hyperplane or strictly convex. Since there do not exist strictly convex surfaces smoothly asymptotic to the flat product \({\Gamma }\times \mathbb {R}\), only the first case will be possible. □
However, one should note that a simple scaling argument shows that selfexpanding hyperplanes are not necessarily equal, if they are asymptotic to each other.
6 Selfexpanders in Higher Codimension
Now we will extend Theorem 1 to the case where \(M^{m}\subset \mathbb {R}^{n}\) is a selfexpander in higher codimension. The idea is to study the same quantity as in [30] for selfshrinkers. Let A^{H} = 〈A, H〉 be the second fundamental form with respect to the mean curvature vector H. Instead of considering the quotient A^{2}/H^{2} as in the last chapter, we treat the scaling invariant quotient A^{H}^{2}/H^{4} which for hypersurfaces coincides with A^{2}/H^{2}. As in [30] we will see that this quantity has a much better behavior. In addition, in this section we will always assume that H > 0 and that the principal normal vector field
is parallel in the normal bundle, i.e.,
This condition is redundant for hypersurfaces but turns out to be crucial in the forthcoming computations. Consequently we have
The computations in [30] for selfshrinkers carry over almost unchanged, in particular, Lemma 3.3 in [30] now becomes
Lemma 2
Let \(M^{m}\subset \mathbb {R}^{n}\) be a selfexpander with H > 0 and parallel principal normal ξ. Then the following equation holds.
In the sequel we will need the following operator. Let E, F be two vector bundles over M and suppose C ∈ Γ(E ⊗ T^{∗}M ⊗ T^{∗}M) and D ∈ Γ(F ⊗ T^{∗}M ⊗ T^{∗}M) are two bilinear forms with values in the vector bundles E respectively F. For example C could be the bilinear form A^{H} (in which case E is the trivial bundle) or D could be the second fundamental tensor A ∈ Γ(T^{⊥}M ⊗ T^{∗}M ⊗ T^{∗}M). Then \(C\circledast D\in {\Gamma }(E\otimes F\otimes T^{\ast } M\otimes T^{\ast } M)\) is by definition the bilinear form given by the trace
where \(e_{1},\dots , e_{m}\) is an arbitrary orthonormal frame in TM.
Theorem 2
Let \(M^{m}\subset \mathbb {R}^{n}\) be a complete and connected selfexpander with H≠ 0, bounded second fundamental form A and parallel principal normal ξ = H/H. Then the following statements are equivalent:

(a)
M is a selfexpanding hyperplane \({\Gamma }\times \mathbb {R}^{m1}\).

(b)
The function \(\frac {A^{H}^{2}}{H^{4}}\) attains a local maximum.
If one of these equivalent conditions is satisfied, then \(\frac {A^{H}^{2}}{H^{4}}\) is constant to 1.
Proof
The proof will be separated into several steps.

(i)
First note that
$$ A^{H}\circledast A=A\circledast A^{H}. $$This is a consequence of Ricci’s equation in Proposition 1(b) and of ∇^{⊥}ξ = 0, because
$$ \begin{array}{@{}rcl@{}} 0&=&HR^{\perp}(v,w)\xi=R^{\perp}(v,w)H\\ &=&(A\circledast A^{H}A^{H}\circledast A)(v,w). \end{array} $$ 
(ii)
The strong elliptic maximum principle and (8) imply that
$$ \frac{A^{H}^{2}}{H^{4}}=c $$for some constant c > 0 and
$$ \nablaH\otimes\frac{A^{H}}{H}H\nabla\frac{A^{H}}{H}=0. $$(9)From Codazzi’s equation and since ξ is parallel we obtain that \(\nabla ^{\perp }\frac {A^{H}}{H}=\nabla ^{\perp } A^{\xi }\) is fully symmetric. Then as in [30] we can decompose the quantity on the lefthand side in (9) into its symmetric and antisymmetric parts to derive that ∇H⊗ A^{H} is fully symmetric and therefore
$$ A^{H}^{2}\nablaH^{2}(A^{H}\circledast A^{H})(\nablaH,\nablaH)=0. $$ 
(iii)
We will distinguish two cases.
Case 1. Suppose that ∇H = 0 on M which in view of ∇^{⊥}ξ = 0 is equivalent to ∇^{⊥}H = 0. Then λ > 0 and (4) show that H = 0 which is a contradiction to our assumption (in fact, the same equation shows that on any selfexpander the function H cannot attain local positive minima). So this case cannot occur.
Case 2. From the full symmetry of the tensor ∇H ⊗ A^{H} that we obtained in step (ii) one derives that at a point p ∈ M where ∇H(p)≠ 0 any tangent vector v ∈ T_{p}M orthogonal to ∇H(p) is a zero eigenvector of A^{H} at p and that ∇H is an eigenvector of A^{H} to the eigenvalue H^{2} (since trace(A^{H}) = H^{2}). In particular, the tensor A^{H} has only one nonzero eigenvalue and A^{H}^{2} = H^{4} on all of M. Thus as in [30] on the open set
$$ M^{o}:=\{p\in M:\nabla H(p)\neq 0\} $$we define the two distributions
$$ \begin{array}{@{}rcl@{}} \mathcal{E}_{p}M^{o} &:=& \{v\in T_{p}M^{o}:A^{H}(v,\cdot)=H^{2}\langle v,\cdot\rangle\},\\ \mathcal{F}_{p}M^{o} &:=& \{v\in T_{p}M^{o}:A^{H}(v,\cdot)=0\}. \end{array} $$Taking into account Theorem 3.20 in [10] or Lemma 6.4 in [12], we may then proceed exactly as in [30] to prove that \({\mathcal{E}}\), \({\mathcal{F}}\) can be smoothly extended to parallel distributions on all of M and that M splits into the Riemannian product \(M={\Gamma }\times \mathbb {R}^{m1}\), where Γ is a selfexpanding curve, and that the distributions \({\mathcal{E}}, {\mathcal{F}}\) form the tangent bundles of Γ respectively \(\mathbb {R}^{m1}\).
This completes the proof of Theorem 2. □
Change history
06 May 2021
Funding information was added to the article.
Notes
Most computations will hold as well for λ < 0, i.e., for selfshrinkers.
References
Begley, T., Moore, K.: On short time existence of Lagrangian mean curvature flow. Math. Ann. 367, 1473–1515 (2017)
Bernstein, J., Wang, L.: Smooth compactness for spaces of asymptotically conical selfexpanders of mean curvature flow. Int. Math. Res Not. https://doi.org/10.1093/imrn/rnz087 (2019)
Bernstein, J., Wang, L.: The space of asymptotically conical selfexpanders of mean curvature flow. arXiv:1712.04366 (2017)
Cheng, X., Zhou, D.: Spectral properties and rigidity for selfexpanding solutions of the mean curvature flows. Math. Ann. 371, 371–389 (2018)
Clutterbuck, J., Schnürer, O.C.: Stability of mean convex cones under mean curvature flow. Math. Z. 267, 535–547 (2011)
Ding, Q.: Minimal cones and selfexpanding solutions for mean curvature flows. Math. Ann. 376, 359–405 (2020)
Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. (2) 130, 453–471 (1989)
Fong, F.T.H., McGrath, P.: Rotational symmetry of asymptotically conical mean curvature flow selfexpanders. Commun. Anal. Geom. 27, 599–618 (2019)
Gȧrding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
Groh, K., Schwarz, M., Smoczyk, K., Zehmisch, K.: Mean curvature flow of monotone Lagrangian submanifolds. Math. Z. 257, 295–327 (2007)
Guan, P., Viaclovsky, J., Wang, G.: Some properties of the Schouten tensor and applications to conformal geometry. Trans. Amer. Math. Soc. 355, 925–933 (2003)
Halldorsson, H.P.: Selfsimilar solutions to the curve shortening flow. Trans. Amer. Math. Soc. 364, 5285–5309 (2012)
Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. In: Greene, R.E., Yau, S.T. (eds.) Differential Geometry: Partial Differential Equations on Manifolds, Part 1 (Los Angeles, CA, 1990). Proceedings of Symposia in Pure Mathematics. vol. 54, pp. 175–191, Amer. Math. Soc., Providence, RI (1993)
Imagi, Y., Joyce, D., dos Santos, J.O.: Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in \(\mathbb {C}^{m}\). Duke Math. J. 165, 847–933 (2016)
Ishimura, N.: Curvature evolution of plane curves with prescribed opening angle. Bull. Aust. Math. Soc. 52, 287–296 (1995)
Jost, J.: Riemannian Geometry and Geometric Analysis, 7th ed. Universitext. Springer, Cham (2017)
Jost, J., Xin, Y., Yang, L.: A spherical Bernstein theorem for minimal submanifolds of higher codimension. Calc. Var. Partial Differ. Equ. 57, 166 (2018)
Jost, J., Xin, Y., Yang, L.: The Gauss image of entire graphs of higher codimension and Bernstein type theorems. Calc. Var. Partial Differ. Equ. 47, 711–737 (2013)
Jost, J., Xin, Y.: Bernstein type theorems for higher codimension. Calc. Var. Partial Differ. Equ. 9, 277–296 (1999)
Joyce, D., Lee, Y.I., Tsui, M.P.: Selfsimilar solutions and translating solitons for Lagrangian mean curvature flow. J. Differ. Geom. 84, 127–161 (2010)
Lee, Y.I., Wang, M.T.: Hamiltonian stationary shrinkers and expanders for Lagrangian mean curvature flows. J. Differ. Geom. 83, 27–42 (2009)
Lotay, J.D., Neves, A.: Uniqueness of Langrangian selfexpanders. Geom. Topol. 17, 2689–2729 (2013)
Martín, F., SavasHalilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. Partial Differ. Equ. 54, 2853–2882 (2015)
Nakahara, H.: Some examples of selfsimilar solutions and translating solitons for Lagrangian mean curvature flow. Tohoku Math. J. 65(2), 411–425 (2013)
Rasul, K.: Slow convergence of graphs under mean curvature flow. Commun. Anal. Geom. 18, 987–1008 (2010)
SavasHalilaj, A., Smoczyk, K.: Mean curvature flow of area decreasing maps between Riemann surfaces. Ann. Glob. Anal. Geom. 53, 11–37 (2018)
SavasHalilaj, A., Smoczyk, K.: Evolution of contractions by mean curvature flow. Math. Ann. 361, 725–740 (2015)
SavasHalilaj, A., Smoczyk, K.: Homotopy of area decreasing maps by mean curvature flow. Adv. Math. 255, 455–473 (2014)
SavasHalilaj, A., Smoczyk, K.: Bernstein theorems for length and area decreasing minimal maps. Calc. Var. Partial Differ. Equ. 50, 549–577 (2014)
Smoczyk, K.: Selfshrinkers of the mean curvature flow in arbitrary codimension. Int. Math. Res. Not. 2005, 2983–3004 (2005)
Smoczyk, K., Wang, G., Xin, Y.L.: Bernstein type theorems with flat normal bundle. Calc. Var. Partial Differ. Equ. 26, 57–67 (2006)
Stavrou, N.: Selfsimilar solutions to the mean curvature flow. J. Reine Angew. Math. 499, 189–198 (1998)
Acknowledgements
The author was supported by the German Research Foundation within the priority program SPP 2026  Geometry at Infinity, DFG SM 78/71.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dedicated to my teacher, supervisor and friend Jürgen Jost on the occasion of his 65th birthday.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Smoczyk, K. SelfExpanders of the Mean Curvature Flow. Vietnam J. Math. 49, 433–445 (2021). https://doi.org/10.1007/s10013020004691
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013020004691
Keywords
 Mean curvature flow
 Selfexpander
Mathematics Subject Classification (2010)
 53C44
 53C21
 53C42