Abstract
Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds.
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1 Introduction
Many inequalities that relate the mean curvature of submanifolds with other geometric quantities such as the diameter can be obtained in some way from monotonicity identities, which are formulas that can be used to deduce monotonicity of weighted density ratios. In the Euclidean case, these identities are typically proven by testing the first variation formula with certain vector fields. One of the main ingredients in the construction of these vector fields is the inclusion map of the submanifold into the ambient Euclidean space. A key observation in the computations is that its relative divergence equals the dimension of the submanifold. In the Riemannian case, the inclusion map of a submanifold is not a vector field; however, one can perform analogous arguments by using the vector field \(r \nabla r\), where r is the distance function to a given point (see, for instance, Anderson [3]). Indeed, its relative divergence is not constant but can be bounded below on small geodesic balls by Rauch’s comparison theorem (see Lemma 2.7). Such an idea revealed to be very fruitful; for instance, it enabled Hoffman–Spruck [11] to derive a Sobolev inequality for Riemannian manifolds. The idea of testing the first variation formula with the vector field \(r\nabla r\) in combination with Hessian comparison theorems for the distance function that give a lower bound of the relative divergence was used again in the works of Karcher–Wood [14] and Xin [44]. Their resulting monotonicity inequalities imply Liouville-type vanishing theorems for harmonic vector bundle valued p-forms. Later, the same idea was used by several authors to prove vanishing theorems in various settings, see, for instance, Dong–Wei [8]. The technique was recently applied by Mondino–Spadaro [26] to derive an inequality that relates the radius of balls with the volume and area of the boundary. See also Nardulli–Osorio Acevedo [27] who used the technique to prove monotonicity inequalities for varifolds on Riemannian manifolds. A weighted monotonicity inequality was obtained by Nguyen [28].
In the present paper, we apply the described technique to prove Riemannian counterparts of the Euclidean monotonicity inequalities and their consequences from Simon [38, 39] and Allard [2]. Our main result is a general Li–Yau inequality (see Theorem 1.7). We start with a brief introduction to intrinsic varifolds on Riemannian manifolds in Sect. 1.1. All our monotonicity inequalities (see Sect. 3) as well as our main theorems (see Sect. 1.3) are valid for general varifolds. In particular, all our results can be applied to isometrically immersed Riemannian submanifolds (see Example 2.4). Indeed, the Li–Yau inequality for ambient manifolds with positive upper bound on the sectional curvature is also new in the smooth case.
1.1 Varifolds on Riemannian manifolds
Let m, n be positive integers satisfying \(m\le n\). Given any n-dimensional vector space V, we define the Grassmann manifold \({\mathbf {G}}(V,m)\) to be the set of all m-dimensional linear subspaces of V. For \(V = {\mathbf {R}}^n\), we write \({\mathbf {G}}(n,m):={\mathbf {G}}({\mathbf {R}}^n,m)\). One can show that \({\mathbf {G}}(n,m)\) is a smooth Euclidean submanifold, see for instance [9, 3.2.29(4)].
Let (N, g) be an n-dimensional Riemannian manifold. We denote with \({\mathbf {G}}_m(TN)\) the Grassmann m-plane bundle of the tangent bundle TN of N. That is, there exists a map \(\pi :{\mathbf {G}}_m(TN)\rightarrow N\) such that for each \(p\in N\), the fibre \(\pi ^{-1}(p)\) is given by the Grassmannian manifold \({\mathbf {G}}(T_pN,m)\). Given any open set U in N and a chart \(x:U\rightarrow {\mathbf {R}}^n\) of N, we note that \(\pi ^{-1}[U]\) is homeomorphically mapped onto an open subset of \({\mathbf {R}}^n\times {\mathbf {G}}(n,m)\) via
This turns \({\mathbf {G}}_m(TN)\) into a differentiable manifold. We define
and note that \({\mathbf {G}}_m(N)\) and \({\mathbf {G}}_m(TN)\) are homeomorphic. In particular, \({\mathbf {G}}_m(N)\) is a locally compact and separable metric space.
With an m-dimensional varifold in N we mean a Radon measure V over \({\mathbf {G}}_m(N)\). The space of all m-dimensional varifolds on N is denoted with \({\mathbf {V}}_m(N)\). The weight measure \(\Vert V\Vert \) of a varifold V is defined by
It is the push forward measure of the varifold under the projection \({\mathbf {G}}_m(N)\rightarrow N\). In particular, \(\Vert V\Vert \) is a Radon measure on N (see [22, Lemma 2.6]).
The space of compactly supported vector fields on N is denoted with \({\mathscr {X}}(N)\). Given any \(X\in {\mathscr {X}}(N)\), \(p\in N\), and \(T\in {\mathbf {G}}(T_pN,m)\) with orthonormal basis \(\{e_1,\ldots ,e_m\}\), we let
where \(\nabla \) denotes the Levi-Civita connection. Moreover, we denote with \({{\,\mathrm{spt}\,}}X\) the support of X. The first variation of a varifold V is defined as the linear functional
The total variation \(\Vert \delta V\Vert \) of \(\delta V\) is defined by
whenever U is an open subset of N, and
whenever A is any subset of N.
Finally, we say that H is the generalised mean curvature of V in (N, g), if and only if \(H:N\rightarrow TN\) is \(\Vert V\Vert \) measurable, \(\Vert \delta V\Vert \) is a Radon measure over N, there exists a \(\Vert \delta V\Vert \) measurable map \(\eta \) taking values in TN such that \(\Vert \delta V\Vert \) almost everywhere, \(g(\eta ,\eta )\le 1\), and
where \(\Vert \delta V\Vert _{\mathrm {sing}} = \Vert \delta V\Vert - \Vert \delta V\Vert _{\Vert V\Vert }\), and \(\Vert \delta V\Vert _{\Vert V\Vert }\) is the absolutely continuous part of \(\Vert \delta V\Vert \) with respect to \(\Vert V\Vert \), see [9, 2.9.1].
It remains to mention that each isometrically immersed Riemannian manifold \(M\rightarrow N\) can be considered as a varifold in N. For more details, see Example 2.4.
1.2 Notation and definitions
Suppose (X, d) is a metric space, \(\mu \) is a Radon measure on X, and m is a positive integer. Denote with \(\varvec{\alpha }(m)\) the volume of the unit ball in \({\mathbf {R}}^m\). Given any \(p\in X\) and \(r>0\), we define the balls
The m-dimensional lower density \( {\varvec{\Theta }}^{m}_*(\mu ,p)\) and upper density \( {\varvec{\Theta }}^{*m}(\mu ,p)\) of \(\mu \) at \(p\in X\) are defined by
Moreover, if \({\varvec{\Theta }}^{m}_*(\mu ,p)= {\varvec{\Theta }}^{*m}(\mu ,p)\), we let \(\varvec{\Theta }^{m}(\mu ,p):= {\varvec{\Theta }}^{m}_*(\mu ,p)\). The support \({{\,\mathrm{spt}\,}}\mu \) of the measure \(\mu \) is defined by
1.1 Definition
Suppose N is a Riemannian manifold and \(p\in N\).
We say that U is an open geodesically star-shaped neighbourhood of p if and only if there exists an open star-shaped neighbourhood D of 0 in \(T_pN\) such that the exponential map \(\exp _p:D\rightarrow U\) is a diffeomorphism with \(\exp _p[D] = U\), and all geodesics emanating from p are length-minimising in U.
Similarly, we say that the open ball \(B_r(p)\) with radius \(r>0\) is a geodesic ball if it is a geodesically star-shaped neighbourhood of p.
Given a complete Riemannian manifold N, \(p\in N\), and writing \(r=d(p,\cdot )\), we denote with \({{\,\mathrm{Cut}\,}}(p)\) the cut locus of p in N, and define the radial curvature \(K_r\) on \(N{\setminus }{{\,\mathrm{Cut}\,}}(p)\) to be the restriction of the sectional curvature to all planes that contain the gradient \(\nabla r\) of r. Moreover, given any subset \(A\subset N\), we denote with \(i_p(A)\) the injectivity radius at p in N restricted to the subset A.
Typically, we denote with \(|\cdot |_g\) the norm induced by a Riemannian metric g.
1.3 Geometric inequalities
The following lower diameter bound for closed minimal submanifolds was proven and discussed by Xia [43] for the special case where the ambient manifold N is given by the n-dimensional real projective space of curvature 1. To the author’s knowledge, little is known in the general case. In particular, the study of sharp lower bounds for different model spaces remains an open problem. The theorem is a direct consequence of Lemma 4.1 in combination with (2.9) and Example 2.4.
1.2 Theorem
(Lower diameter bounds for closed minimal Riemannian submanifolds) Suppose N is a complete Riemannian manifold, M is a closed minimal submanifold of N with extrinsic diameter \(d_{\mathrm {ext}}(M)=\sup _{M\times M} d\), \(b>0\), the sectional curvature K of N satisfies \(\sup _MK\le b\), and \(p\in M\).
Then, there holds
1.3 Remark
Notice that if the sectional curvature K of N satisfies \(K\le 0\), then (1.2) becomes
If instead, the sectional curvatures of N are pinched between \(\frac{1}{4}b\) and b, and N is simply connected, then, by a result of Klingenberg [16], (1.2) becomes
In the special case where \(N=\mathbf{RP}^n\) is the n-dimensional real projective space of curvature 1, the lower bound in (1.4) is attained if and only if M is totally geodesic, see [43]. However, the same does not hold for the unit sphere \(N={\mathbf {S}}^7\). Indeed, the complex projective space \(\mathbf{CP}^2(\frac{4}{3})\) of complex dimension 2 and complex sectional curvature \(\frac{4}{3}\) has diameter \(\frac{\sqrt{3}\pi }{2}\) and can be isometrically and minimally imbedded into \({\mathbf {S}}^7\), see [34, 43]. It remains an interesting open problem to study sharp lower diameter bounds for closed minimal submanifolds in different ambient manifolds, in particular in \({\mathbf {S}}^n\).
We have the following generalisation of (1.3) to asymptotically non-positively curved ambient manifolds as a consequence of Lemma 4.3 in combination with (2.9), and Example 2.4.
1.4 Theorem
Suppose m, n are positive integers, \(m\le n\), N is a complete n-dimensional Riemannian manifold, \(p\in N\), \(r = d(p,\cdot )\), \(D(p) := N{\setminus }({{\,\mathrm{Cut}\,}}(p)\cup \{p\})\), \(0<b\le 1/4\), the radial curvature \(K_r\) satisfies \(K_r \le \frac{b}{r^2}\) on D(p), and M is a closed minimal submanifold of N with \(p\in M\).
Then, the extrinsic diameter \(d_{\mathrm {ext}}(M)\) of M in N can be bounded below by the cut locus distance:
In [39, Lemma 1.1], Simon showed a diameter pinching theorem for closed surfaces in the Euclidean space in terms of their Willmore energy and their area. The upper diameter bound was improved and generalised to Euclidean submanifolds by Topping [40] using the Michael–Simon Sobolev inequality [24]. It was further generalised to Euclidean submanifolds with boundary by Menne–Scharrer [22] leading to a priori diameter bounds for solutions of Plateau’s problem. The Riemannian equivalent of Topping’s upper diameter bound was proven by Wu–Zheng [42] using the Hoffman–Spruck Sobolev inequality [11]. The following theorem is the Riemannian version for varifolds of Simon’s diameter pinching. It is proven in Sect. 4.1. In the smooth case, it follows from [42].
1.5 Theorem
[Diameter pinching] Suppose n is an integer, \(n\ge 2\), N is a complete n-dimensional Riemannian manifold with positive injectivity radius \(i>0\), \(b\ge 0\), the sectional curvature satisfies \(K\le b\), \(V\in {\mathbf {V}}_2(N)\) has generalised mean curvature H, H is square integrable with respect to \(\Vert V\Vert \), \(\Vert \delta V\Vert \) is absolutely continuous with respect to \(\Vert V\Vert \), \({{\,\mathrm{spt}\,}}\Vert V\Vert \) is compact and connected,
and
If \(\Vert V\Vert (N)\le C_{(4.2)}(i,b)\), then
and the extrinsic diameter \(d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )\) of the support of \(\Vert V\Vert \) is bounded above:
Moreover, if \(d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )<\min \{i,\frac{\pi }{3\sqrt{b}}\}\), then
1.6 Remark
Notice that if the injectivity radius is infinite and the sectional curvatures are non-positive, then both the condition on the area for the upper bound and the condition on the diameter for the lower bound disappear. Indeed, for \(i=\infty \) and \(b=0\), there holds \(C_{4.2}(i,b) = \infty = \min \{i,\frac{\pi }{3\sqrt{b}}\}\). Notice also that if \(b=0\), then the diameter bounds are the exact analogous of Simon’s Euclidean version [39, Lemma 1.1]. In fact, the only difference is the restriction on the area through the injectivity radius in case it is not infinite.
If \(N = {\mathbf {R}}^n\), then the condition on the generalised mean curvature to be normal (1.5) is satisfied for all integral varifolds, see [4, Sect. 5.8]. Similarly, condition (1.6) is satisfied for all integral varifolds. On the other hand, (1.6) implies rectifiability, see [2, Theorem 5.5(1)].
In the early 60s, Willmore [41] showed that the energy now bearing his name is bounded below by \(4\pi \) on the class of closed surfaces \(\Sigma \subset {\mathbf {R}}^3\):
where H denotes the sum of the principal curvatures and \(\mu \) is the canonical Radon measure on \(\Sigma \) given by the immersion. The inequality is also referred to as Willmore inequality. Equality holds only for round spheres. Willmore’s inequality was improved by Li–Yau [19, Theorem 6] for smoothly immersed closed surfaces \(f:\Sigma \rightarrow {\mathbf {R}}^n\): If there exists \(p\in {\mathbf {R}}^n\) with \(f^{-1}(p)=\{x_1,\ldots ,x_k\}\) where the \(x_i\)’s are all distinct points in \(\Sigma \), in other words f has a point of multiplicity k, then
In particular, if the Willmore energy \(\frac{1}{4}\int _{\Sigma }|H|^2\,\mathrm d\mu \) lies strictly below \(8\pi \), then f is an embedding. Because of this property, the Li–Yau inequality has become very useful for the minimisation of the Willmore functional and, more generally, for the study of immersed surfaces. Due to the conformal invariance of the Willmore functional observed by Chen [7], Willmore’s inequality has an analogue for surfaces \(\Sigma \) in the three sphere \({\mathbf {S}}^3\):
and an analogue for surfaces \(\Sigma \) in the hyperbolic space \({\mathbf {H}}^3\):
where \(|\Sigma |=\int 1\,\mathrm d\mu \) denotes the area of \(\Sigma \). Kleiner [15] showed
for minimisers \(\Sigma _0\) of the isoperimetric profile in a complete one-connected three-dimensional Riemannian manifold without boundary and with sectional curvatures bounded above by \(b\le 0\). Then, Ritoré [30] showed that (1.12) remains valid for all \(C^{1,1}\) surfaces in a three-dimensional Cartan–Hadamard manifold with sectional curvatures bounded above by \(-1\). Schulze showed that the classical Willmore inequality (1.10) holds true for integral 2-varifolds in three-dimensional Cartan–Hadamard manifolds [36, Lemma 6.7]. Then, he showed that (1.13) remains valid for integral 2-varifolds in n-dimensional Cartan–Hadamard manifolds, see [37, Theorem 1.4]. Finally, Chai [5] upgraded (1.12) to a Li–Yau inequality. That is, given a smoothly immersed surface \(f:\Sigma \rightarrow {\mathbf {H}}^n\) in the hyperbolic space \({\mathbf {H}}^n\) which has a point of multiplicity k, then
Notice also the recent generalisation of the Willmore inequality to higher-dimensional submanifolds by Agostiniani–Fogagnolo–Mazzieri [1]. They showed that for closed codimension 1 submanifolds M in a non-compact n-dimensional Riemannian manifold (N, g) with nonnegative Ricci curvature, there holds
where \(\mathrm {AVG}(g)\) denotes the asymptotic volume ratio of (N, g). See also Chen [6] for the earlier Euclidean version.
In our following theorem, we upgrade the result of Schulze [37] about the inequality (1.13) to a Li–Yau inequality in any non-positively curved ambient manifold. More importantly, we upgrade the spherical Willmore inequality (1.11) to a Li–Yau inequality for varifolds on any Riemannian manifold with an upper bound (possibly positive) on the sectional curvature. The proof is done in Sect. 3.1. For non-positively curved ambient manifolds, it is analogous to Chai [5]; for ambient manifolds with positive upper bound on the curvature it is inspired by Simon [39].
1.7 Theorem
(Li–Yau inequality) Suppose n is an integer, \(n\ge 2\), (N, g) is an n-dimensional Riemannian manifold, \(p\in N\), U is a geodesically star-shaped open neighbourhood of p, \(V\in {\mathbf {V}}_2(N)\) has generalised mean curvature H, H is square integrable with respect to \(\Vert V\Vert \), \(H(x)\bot T\) for V almost all \((x,T) \in {\mathbf {G}}_2(N)\), \(p\notin {{\,\mathrm{spt}\,}}\Vert \delta V\Vert _{\mathrm {sing}}\), \({{\,\mathrm{spt}\,}}\Vert V\Vert \) is compact, \({{\,\mathrm{spt}\,}}\Vert V\Vert \subset U\), \(b \in {\mathbf {R}}\), the sectional curvature of N satisfies \(\sup _{{{\,\mathrm{spt}\,}}\Vert V\Vert }K\le b\), and either
or
Then, there holds
where \(t_b(r) = 2/r\) if \(b\ge 0\), and \(t_b(r) = 2\sqrt{|b|}\coth (\sqrt{|b|}r/2)\) if \(b<0\).
1.8 Remark
Notice that the existence of the density \({\varvec{\Theta }}^2(\Vert V\Vert ,p)\) is part of the statement. Indeed, existence of the density as well as its upper semi-continuity is local statements that do not require any global upper bounds on the curvature nor do they require positive injectivity radius, see Theorem 3.6.
If the varifold is given by a smoothly immersed surface, then the theorem reads as follows (see Example 2.4).
1.9 Corollary
Suppose n is an integer, \(n\ge 3\), (N, g) is an n-dimensional Riemannian manifold, \(b\in {\mathbf {R}}\), the sectional curvature K of N satisfies \(K\le b\), \(\Sigma \) is a closed surface, \(f:\Sigma \rightarrow N\) is a smooth immersion, and f has a point of multiplicity k. Let H be the trace of the second fundamental form of the immersion f, \(\mu \) be the Radon measure on \(\Sigma \) induced by the pull-back metric of g along f, and \(|\Sigma | := \int _{\Sigma }1\,\mathrm d\mu \) be the area of \(\Sigma \) in N. Then, the following two statements hold.
-
1.
If \(b>0\), and the image of f is contained in a geodesic ball of radius at most \(\frac{\pi }{2\sqrt{b}}\), then
$$\begin{aligned} \frac{1}{4}\int _{\Sigma }|H|^2_g\,\mathrm d\mu + \frac{16}{\pi ^2}b|\Sigma |\ge 4\pi k. \end{aligned}$$ -
2.
If \(b\le 0\) and the image of f is contained in a geodesically star-shaped open neighbourhood of the multiplicity point, then
$$\begin{aligned} \frac{1}{4}\int _{\Sigma }|H|^2_g\,\mathrm d\mu + b|\Sigma |\ge 4\pi k. \end{aligned}$$
In particular, if the left-hand side is strictly smaller than \(8\pi \), then f is an embedding.
1.10 Remark
The upper bound on the radius of the geodesic ball containing the image of f in (1) can be enlarged up to \(\frac{\pi }{\sqrt{b}}\) at the cost of a constant larger than \(\frac{16}{\pi ^2}\) in front of the area. Moreover, in view of Lemma 3.1 and (3.5), the constant \(\frac{16}{\pi ^2}\) cannot be expected to be sharp. It is an interesting open question whether or not the constant \(\frac{16}{\pi ^2}\) can be replaced by 1. In view of the spherical version (1.11), this seems possible.
If N is a Cartan–Hadamard manifold, then N itself is a geodesically star-shaped open neighbourhood of any point. In particular, there is no condition on f in (2).
Isoperimetric inequalities play an important role in the theory of varifolds and its applications, see, for instance, [2, 20, 22]. They can be derived from Sobolev inequalities. Allard [2, Theorem 7.1] proved a Sobolev inequality for general varifolds in Euclidean space with a constant depending on the dimension of the varifold and the dimension of the ambient space. Michael–Simon [24] proved a Sobolev inequality for generalised submanifolds in Euclidean space where the constant depends only on the dimension of the submanifold. Its proof was later adapted by Simon [38, Theorem 18.6] for varifolds whose first variation is absolutely continuous with respect to the weight measure. These varifolds correspond to submanifolds without boundary. Menne–Scharrer [23] proved a general Sobolev inequality accounting for the unrectifiable part of the varifold as well. Finally, Hoyos [12, 13] generalised Simon’s version [38, Theorem 18.6] for varifolds on Riemannian manifolds. Our following theorem generalises his inequality for arbitrary varifolds whose first variation doesn’t have to be absolutely continuous with respect to the weight measure. In this way, our resulting isoperimetric inequality indeed recovers the smooth version of Hoffman–Spruck [11, Theorem 2.2]. The proof of the following theorem can be found in Sect. 5.1.
1.11 Theorem
(Sobolev inequality) Suppose m, n are positive integers, \(m\le n\), (N, g) is a complete n-dimensional Riemannian manifold, \(V\in {\mathbf {V}}_m(N)\), \(\Vert \delta V\Vert \) is a Radon measure, \(i({{\,\mathrm{spt}\,}}\Vert V\Vert )\) denotes the injectivity radius of N restricted to \({{\,\mathrm{spt}\,}}\Vert V\Vert \), \(b>0\), the sectional curvature K of N satisfies \(\sup _{{{\,\mathrm{spt}\,}}\Vert V\Vert }K\le b\), \(d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )\) denotes the extrinsic diameter of \({{\,\mathrm{spt}\,}}\Vert V\Vert \) in N, and
Then, for all nonnegative compactly supported \(C^1\) functions \(h\le 1\) on N, there holds
where \((\nabla ^\top h)(x,T)\) is the orthogonal projection of \(\nabla h(x)\) onto T for \((x,T)\in {\mathbf {G}}_m(N)\).
One can absorb the curvature-dependent term in the middle to obtain the following isoperimetric inequality. For its proof, see Sect. 5.2.
1.12 Corollary
(Isoperimetric inequality) If in addition
and \({\varvec{\Theta }}^{m}(\Vert V\Vert ,x)\ge 1\) for \(\Vert V\Vert \) almost all \(x\in {{\,\mathrm{spt}\,}}\Vert V\Vert \), then
1.13 Remark
Notice that by Example 2.4, the inequality above indeed implies the isoperimetric inequality [11, Theorem 2.2] for smoothly immersed Riemannian manifolds. Moreover, if the sectional curvatures of N are non-positive (for instance, if N is Cartan–Hadamard), then the condition on the volume (1.14) is satisfied for all V with compact support.
2 Preliminaries
In this section, we show how the first variation can be represented by integration, see Lemmas 2.1 and 2.2. These are basic facts that have been frequently used in the study of Euclidean varifolds. We are going to need them in order to prove the Sobolev inequality (Theorem 1.11). Moreover, we prove that any smoothly immersed manifold is a varifold, see Lemma 2.3 and Example 2.4. Finally, we state the Hessian comparison theorems for the distance function (see Lemmas 2.5 and 2.7) that are crucial to derive the monotonicity inequalities in Sect. 3.
2.1 Lemma
(Compare [2, 4.3(2)]) Suppose m, n are positive integers, \(m \le n\), (N, g) is an n-dimensional Riemannian manifold, \(V\in {\mathbf {V}}_m(N)\), and \(\Vert \delta V\Vert \) is a Radon measure.
Then, there exists a \(\Vert \delta V\Vert \) measurable map \(\eta :N\rightarrow TN\) such that \(\Vert \delta V\Vert \) almost everywhere, \(g(\eta ,\eta )\le 1\), and
Remark
Notice that by definition, \(\Vert \delta V\Vert \) is a Borel regular measure on N. Hence, it is a Radon measure if and only if \(\Vert \delta V\Vert (K)<\infty \) for all compact sets \(K \subset N\).
Proof
Let \(\{(x_\alpha ,U_\alpha ):\alpha \in I\}\) be a countable differentiable atlas on N. For all \(\alpha \in I\) define
where we’ve use the Einstein summation convention, and \(\frac{\partial }{\partial x_\alpha ^i}\) are the coordinate vector fields corresponding to the chart \(x_\alpha \). Then,
Therefore, we can apply the representation theorem [9, 2.5.12] in combination with Lusin’s theorem [9, 2.3.6] to obtain a Borel map \(k_\alpha \) on \(U_\alpha \) taking values in the dual space \(({\mathbf {R}}^n)^*\) of \({\mathbf {R}}^n\) such that \(\Vert \delta V\Vert \) almost everywhere, \(k_{\alpha i}k_{\alpha j}g_\alpha ^{ij} = 1\), and
For \(\eta _\alpha := k_{\alpha i}g_\alpha ^{ij}\frac{\partial }{\partial x_\alpha ^j}\), this means that \(\Vert \delta V\Vert \) almost everywhere, \(g(\eta _\alpha ,\eta _\alpha ) = 1\), and
whenever \(X= X_\alpha ^i\frac{\partial }{\partial x_\alpha ^i} \in {\mathscr {X}}(U_\alpha )\). This equation uniquely determines the Borel map \(\eta _\alpha \) up to a set of \(\Vert \delta V\Vert \) measure zero. Hence, we can define a \(\Vert \delta V\Vert \) measurable map \(\eta :N\rightarrow TN\) by requiring that
Now, let \(X\in {\mathscr {X}}(N)\). Choose a finite covering \(\{U_\alpha :\alpha \in I_0\}\) of the support of X together with a subordinate partition of unity \(\{\varphi _\alpha :\alpha \in I_0\}\). Then,
which finishes the proof. \(\square \)
2.2 Lemma
(Compare [2, 4.3(5)]) Suppose m, n are positive integers, \(m\le n\), N is an n-dimensional Riemannian manifold, \(V\in {\mathbf {V}}_m(N)\), and \(\Vert \delta V\Vert \) is a Radon measure.
Then, V has a generalised mean curvature (see (1.1)).
Proof
First, assume that \({{\,\mathrm{spt}\,}}\Vert \delta V\Vert \) is compact. Then, by [9, 2.8.9] one can apply the usual theory of symmetrical derivation (i.e. [9, 2.8.18, 2.9.7]) to the integral in Lemma 2.1. The general case then follows by covering \({{\,\mathrm{spt}\,}}\Vert \delta V\Vert \) with countably many open sets whose closure is compact. \(\square \)
The following lemma is the Riemannian counterpart of [22, Lemma 2.8].
2.3 Lemma
Suppose m, n are positive integers, \(m\le n\), M is a compact m-dimensional connected differentiable manifold with boundary, (N, g) is an n-dimensional Riemannian manifold, and \(f:M\rightarrow N\) is a smooth proper immersion. Denote with \({\mathcal {H}}^m_g\) the m-dimensional Hausdorff measure on N with respect to the distance induced by the metric g, and denote with \(\mu _{f^*g}\) the Riemannian measure on M corresponding to the pull-back metric \(f^*g\) of g along f.
Then, there holds
for all compactly supported continuous functions \(k:M\rightarrow {\mathbf {R}}\). In particular, the push forward measure \(f_\#\mu _{f^*g}\) of \(\mu _{f^*g}\) under f is a Radon measure on N and satisfies
where \({\mathcal {H}}^0\) denotes the counting measure. Moreover, for all \(x\in f[M{\setminus } \partial M]\), there holds
and for \(f_\#\mu _{f^*g}\) almost all x,
Proof
First, suppose that \(\partial M = \varnothing \), M is a submanifold of N and \(f=i\) is the inclusion map. Denote with \(d_M\) and \(d_N\) the distance functions on \((M,i^*g)\) and (N, g), respectively. Clearly, \(d_M(a,b)\ge d_N(a,b)\) for all \(a,b \in M\). By [9, 3.2.46], \(\mu _{i^*g}\) coincides with the m-dimensional Hausdorff measure \({\mathcal {H}}^m_{i^*g}\) on M corresponding to the distance \(d_M\). Thus, \(i_\#\mu _{i^*g}(S) = {\mathcal {H}}^m_{i^*g}(M\cap S) \ge \mathcal {H}^m_g(M\cap S)\) for all \(S\subset N\). To prove the reverse inequality, let \(\lambda >1\) and \(p\in M\). Choose an open neighbourhood U of p in N together with a submanifold chart \(x:U \rightarrow {\mathbf {R}}^n\), i.e. \(x[M\cap U] = x[U]\cap ({\mathbf {R}}^{m}\times \{0\})\). Composing x with a linear map \({\mathbf {R}}^m\times {\mathbf {R}}^{n-m}\rightarrow {\mathbf {R}}^m\times {\mathbf {R}}^{n-m}\), we may assume that \(\mathrm dx_p\) maps an orthonormal basis of \(T_pN\) onto an orthonormal basis of \({\mathbf {R}}^n\). In particular, \(\Vert \mathrm dx_p\Vert = \Vert \mathrm dx^{-1}_{x(p)}\Vert = 1\). Hence, there exists \(\rho >0\) such that \(B_\rho (p) := \{q\in N:d_N(p,q)<\rho \} \subset U\), as well as \(\Vert \mathrm dx|_{B_\rho (p)}\Vert \le \sqrt{\lambda }\) and \(\Vert \mathrm d(x|_{B_\rho (p)})^{-1}\Vert \le \sqrt{\lambda }\). Thus, \(x|_{B_\rho (p)}\) is Lipschitz continuous with Lipschitz constant bounded above by \(\sqrt{\lambda }\). Next, choose \(\rho _0>0\) such that \(\rho _0<\rho \) and the convex hull of \(x[M\cap B_{\rho _0}(p)]\) in \({\mathbf {R}}^m\times \{0\}\) is contained in \(x[M\cap B_\rho (p)]\). Given any \(a,b \in M\cap B_{\rho _0}(p)\), let
Then, \(c:=x^{-1}\circ \gamma \) is a smooth curve in \(M\cap B_\rho (p)\), connecting a with b. Therefore,
This implies \({\mathcal {H}}^m_{i^*g}(M\cap S) \le \lambda ^m\mathcal {H}^m_g(M\cap S)\) for all \(S\subset B_{\rho _0}(p)\). Thus,
in the sense that the left-hand side exists if and only if the right-hand side exists in which case both sides coincide. Moreover, it follows \(i_\#\mu _{i^*g}(B) = {\mathcal {H}}^m_{i^*g}(M\cap B) = \mathcal {H}^m_g(M\cap B)\) for all Borel sets \(B\subset N\), which proves (2.2) for the special case.
Next, suppose that f is an embedding and \(\partial M =\varnothing \). Then, f[M] is a submanifold of N. Denote with \(i:f[M]\rightarrow N\) the inclusion map. Then, \(f:(M,f^*(i^*g)) \rightarrow (f[M],i^*g)\) is an isometry. This means \(f_\#\mu _{f^*(i^*g)} = \mu _{i^*g}\) and, by the first case,
Hence, (2.2) is valid if \(\partial M=\varnothing \) and f is an embedding. Moreover, in this case, (2.3) follows from [33, Chapter II, Corollary 5.5] in combination with Eq. (2.5).
Now, suppose that f is an immersion and \(\partial M = \varnothing \). Let \(k:M\rightarrow {\mathbf {R}}\) be a continuous function with compact support \({{\,\mathrm{spt}\,}}k\) and choose finitely many open sets \(U_1,\ldots ,U_\Lambda \) whose union contains \({{\,\mathrm{spt}\,}}k\) such that \(f|_{U_\lambda }\) is an embedding for \(\lambda = 1,\ldots ,\Lambda \). Pick a subordinate partition of unity \(\{\varphi _\lambda \}_{\lambda =1}^\Lambda \), i.e. \(\sum _{\lambda =1}^\Lambda \varphi _\lambda (p)=1\) for all \(p\in {{\,\mathrm{spt}\,}}k\), and \({{\,\mathrm{spt}\,}}\varphi _\lambda \subset U_\lambda \) for \(\lambda =1,\ldots ,\Lambda \). Given any \(x\in f[M]\), then, by injectivity of \(f|_{U_\lambda }\) for \(\lambda =1,\ldots ,\Lambda \), the union
is disjoint. In particular, denoting with \(\chi _A\) the characteristic function of any given set A,
Hence, using (2.2) for the special case,
This proves (2.1) which readily implies (2.2). It remains to mention that if \(\partial M \ne \varnothing \), then we have that \(\partial (\partial M) = \varnothing \) and \(f|_{\partial M}\) is an immersion. By the first cases it follows that \({\mathcal {H}}^{m-1}_g(f[\partial M]\cap K) < \infty \) for all compact sets \(K\subset N\). Thus, \({\mathcal {H}}^m_g(f[\partial M]) = 0\).
To prove (2.3), we first assume that f is an embedding. Then, the statement follows from [33, Chapter II, Corollary 5.5] in combination with Eq. (2.5). If f is an immersion, let \(x\in f[M{\setminus }\partial M]\) and let \(p_1,\ldots ,p_k\) be distinct points such that \(f^{-1}\{x\}=\{p_1,\ldots ,p_k\}\). Choose pairwise disjoint open sets \(U_1,\ldots ,U_k\) such that for \(i=1,\ldots ,k\) there holds \(p_i\in U_i\). Then, for small \(r>0\), there holds
Hence, (2.3) follows from the special case by linearity of the limit operator.
To prove (2.4), suppose \(x\in f[M{\setminus } \partial M]\), \(p_1,p_2 \in f^{-1}\{x\}\), and \(U_1,U_2\) are disjoint open neighbourhoods of \(p_1,p_2\) in M, respectively, such that \(f|_{U_1},f|_{U_2}\) are embeddings. Assume that \(\mathrm df_{p_1}[T_{p_1}M] \ne \mathrm df_{p_2}[T_{p_2}M]\). Then, we can pick a unit vector \(v_1\in T_xN\) such that \(v_1\in \mathrm df_{p_1}[T_{p_1}M]{\setminus }\mathrm df_{p_2}[T_{p_2}M]\) and there exists \(0<\varepsilon <1\) such that the cone
satisfies \(C\cap \mathrm df_{p_2}[T_{p_2}M] = \varnothing \). Next, we pick \(\varepsilon _1>0\) such that \(\exp _{p_1}:B_{\varepsilon _1}\subset T_{p_1}M\rightarrow M\) is a diffeomorphism on \(B_{\varepsilon _1}:=\{\xi \in T_{p_1}M:(f^*g)_{p_1}(\xi ,\xi )<\varepsilon _1^2\}\) and introduce polar coordinates
where \(S^{m-1} := \{\xi \in T_{p_1}M:(f^*g)_{p_1}(\xi ,\xi )=1\}\), as well as the density function
By [33, Chapter II, Lemma 5.4], there holds \(\mu _{\Theta ^*(f^*g)} = \theta \mu _{g_0}\), where \(g_0\) is the canonical product metric on \((0,\varepsilon _1)\times S^{m-1}\). Hence, for \(E:= (\Theta \circ (\mathrm df_{p_1})^{-1})[C]\) and \(u_1:=(\mathrm df_{p_1})^{-1}(v_1)\), we have by Fubini’s theorem
for all \(0<\rho <\varepsilon _1\). Noting that \(\theta (t,u) = t^{m-1} + O(t^{m+1})\) as \(t\rightarrow 0+\), it follows
where \(\mu _{f^*g}\llcorner E\) denotes the Radon measure on M given by \((\mu _{f^*g}\llcorner E)(B) = \mu _{f^*g}(E\cap B)\) for all Borel sets \(B\subset M\). Hence, by (2.2) applied to \(f|_{U_1}\), there holds \( {\varvec{\Theta }}^m_*({\mathcal {H}}^m_g\llcorner f[E],x)>0\). Notice that if \(\gamma \) is a curve in \(E\cup \{p_1\}\) with \(\gamma (0)=p_1\), then \((f\circ \gamma )^\cdot (0)\in C\cup \{0\}\). Moreover, we make the following observation. Choose \(\rho >0\) such that \(B_\rho (x)\) is a geodesic ball around x in N. Given any unit vector \(v\in T_xN\) and \(\delta >0\), we denote with \(C(v,\delta )\) the image of the set
under the exponential map \(\exp _x:B_\rho (0)\cap T_xN\rightarrow N\). Then, given any smooth curve \(\gamma \) in N with
one can use normal coordinates and differentiability of \(\gamma \) to show that for some \(t_0>0\), there holds
This observation together with compactness of \({{\bar{B}}}_\rho (x)\) can be used to show that for small \(\rho >0\), \(f[E]\cap f[U_2]\cap B_\rho (x)=\varnothing \). Thus, \( {\varvec{\Theta }}^m_*({\mathcal {H}}^m_g\llcorner f[M],x)>1\). Now, the conclusion follows from [9, 2.10.19(5)]. \(\square \)
The following example is the Riemannian counterpart to the Euclidean case [22, Definition 2.14]. Compare also with [17, Sect. 2.2], where smoothness of f is replaced with \(W^{2,2}\)-regularity.
2.4 Example
Let f, M, N be as in Lemma 2.3. Define \(V\in {\mathbf {V}}_m(N)\) by letting
for all continuous functions \(k:{\mathbf {G}}_m(N)\rightarrow {\mathbf {R}}\) with compact support. In view of Lemma 2.3, we have
In particular, \({{\,\mathrm{spt}\,}}\Vert V\Vert = f[M]\) if M is closed. Moreover, for all \(x\in f[M{\setminus }\partial M]\),
and
Identify \(\mathrm df_p[T_pM]\) with \(T_pM\). Let NM be the normal bundle of the immersion f. That is, there exists \(\pi :NM\rightarrow M\) such that for each \(p\in M\), the fibre \(\pi ^{-1}(p)\) is given by the orthogonal complement of \(T_pM\) in \(T_{f(p)}N\). Denote with \(H_f:M\rightarrow NM\) the mean curvature vector field of f, i.e. the trace of the second fundamental form (see [11, Definition 3.1]). Define the \(\Vert V\Vert \) measurable map \(H:N\rightarrow TN\) by
Let \(X\in {\mathscr {X}}(N)\). By a simple computation (see [11, Lemma 3.2(i)]),
whenever \(p\in M\) and \(f(p)=x\), where \((X\circ f)^t\) denotes the orthogonal projection of \((X\circ f)\) onto the tangent bundle TM. Integrating this equation and using Lemma 2.3 as well as the usual Divergence Theorem on M (see [33, Chapter II, Theorem 5.11]), we infer
where \(\nu \) is the outward unit normal vector field on \(\partial M\). In particular, V has generalised mean curvature H,
for all Borel sets \(B\subset N\), and by (2.4),
By definition of H, we have trivially
and
2.5 Lemma
(See [29, Theorem 6.4.3]) Suppose (N, g) is a Riemannian manifold, \(p\in N\), U is a geodesically star-shaped open neighbourhood of p, the metric is represented in geodesic polar coordinates \(g = \mathrm dr\otimes \mathrm dr + g_r\) on U, \(b\in {\mathbf {R}}\), the sectional curvature satisfies \(K\le b\) on U, and either \(b\le 0\) or \(U\subset B_{\frac{\pi }{\sqrt{b}}}(p)\).
Then, the Hessian \(\nabla ^2r\) of r can be bounded below on U by
2.6 Remark
Define the continuous function
where \(a(0)=1\). Using the series expansion
where all higher-order terms are negative, we see that a is strictly decreasing. In particular, since \(\cot (\frac{\pi }{3})=\frac{1}{\sqrt{3}}\), we have
2.7 Lemma
Suppose m, n are positive integers, \(m\le n\), (N, g) is an n-dimensional Riemannian manifold, \(p \in N\), U is a geodesically star-shaped open neighbourhood of p, \(b>0\), the sectional curvature satisfies \(K\le b\) on U, and \(U\subset B_{\frac{\pi }{\sqrt{b}}}(p)\).
Then, writing \(r = d(p,\cdot )\), there holds
for all \(T\in {\mathbf {G}}_m(TU)\).
Proof
Writing the metric in polar coordinates \(g = \mathrm dr\otimes \mathrm dr + g_r\) and using Lemma 2.5 in combination with (2.7), we compute for \(b>0\)
Given any \(T\in {\mathbf {G}}_m(TU)\) with orthonormal basis \(\{e_1,\ldots ,e_m\}\), it follows
which concludes the proof. \(\square \)
3 Monotonicity inequalities
In this section, we prove several monotonicity inequalities. The version for two-dimensional varifolds on general Riemannian manifolds (Lemma 3.1) will be used to prove existence and upper semi-continuity of the density (see Theorem 3.6 in this section), the diameter pinching (see Theorem 1.5 and Sect. 4.1 for its proof), and part (1) of the Li–Yau inequality (see Theorem 1.7 and Sect. 3.1 for its proof). The version for two-dimensional varifolds on non-positively curved manifolds (Lemma 3.7) will be used to prove part (2) of the Li–Yau inequality. The version for general m-dimensional varifolds (Lemma 3.3) is needed to prove the Sobolev inequality (see Theorem 1.11 and Sect. 5.1 for its proof).
The proof of the following lemma is based on the ideas of the monotonicity formula in Simon [39] in combination with a technique of Anderson [3]. See also [31, Lemma A.3] for a proof in the presence of boundary, and [25] for higher-dimensional varifolds.
3.1 Lemma
Suppose n is an integer, \(n\ge 2\), (N, g) is an n-dimensional Riemannian manifold, \(p\in N\), \(V\in {\mathbf {V}}_2(N)\) has generalised mean curvature H, H is square integrable with respect to \(\Vert V\Vert \), \(H(x)\bot T\) for V almost all \((x,T)\in {\mathbf {G}}_2(N)\), \(b>0\), \(0<\rho <\frac{\pi }{\sqrt{b}}\), the sectional curvature satisfies \(K\le b\) on \({{\,\mathrm{spt}\,}}\Vert V\Vert \cap B_\rho (p)\), U is a geodesically star-shaped open neighbourhood of p, and \({{\,\mathrm{spt}\,}}\Vert V\Vert \cap {{\bar{B}}}_\rho (p) \subset U\).
Then, writing \(r=d(p,\cdot )\), there holds
for all \(0<\sigma <\rho \), where \(a_b(r) = \sqrt{b}r\cot (\sqrt{b}r)\).
Proof
Given any \(\sigma< t < \rho \) and any nonnegative smooth function \(\varphi :{\mathbf {R}}\rightarrow {\mathbf {R}}\) whose support is contained in the interval \((-\infty ,1)\), we let \(X = \varphi (\frac{r}{t})r\nabla r\) and compute
for all \(T\in {\mathbf {G}}_2(TN)\), where \(\nabla ^Tr\) denotes the orthogonal projection of \(\nabla r\) onto T. We write
and notice that
Therefore, testing the first variation equation [see (1.1)] with X, we infer by Lemma 2.7
There holds
Hence, adding \(\int _N2\varphi (\frac{r}{t})(1 - a_b(r))\,\mathrm d\Vert V\Vert \) on both sides of the inequality and multiplying with \(\frac{1}{t^3}\), it follows
Given any \(0<\lambda < 1\), choose \(\varphi \) such that \({{\dot{\varphi }}} \le 0\), \(\varphi (s)=1\) for all \(s\le \lambda \), and \(\varphi (s) = 0\) for all \(s\ge 1\). In other words, \(\varphi \) approaches the characteristic function of the open interval \((-\infty ,1)\) from below as \(\lambda \rightarrow 1\). In particular, if \({{\dot{\varphi }}}(\frac{r}{t}) \ne 0\), then \(\frac{r}{t}\ge \lambda \). Hence,
Moreover, given any \(\Vert V\Vert \) integrable real-valued function f, one computes using Fubini’s theorem, writing \(r_\sigma := \max \{\sigma ,r\}\), and denoting with \(\chi _A\) the characteristic function of any set A,
Therefore, putting (3.2) into (3.1), integrating with respect to t from \(\sigma \) to \(\rho \), letting \(\lambda \rightarrow 1\), and using (3.3), we infer
where \(\pi :{\mathbf {G}}_2(N) \rightarrow N\) denotes the canonical projection. Observe that
Thus, it follows
which, in view of (2.7), implies the conclusion. \(\square \)
3.2 Remark
In the above proof, we let \(\varphi \) approach the characteristic function of the interval \((-\infty ,1)\) from below. If we instead let \(\varphi \) approach the characteristic function of the interval \((-\infty ,1]\) from above, we obtain the following version for closed balls:
for all \(0<\sigma <\rho \).
Define the function
Then, using the series expansion of \(\cot (x)\) [see (2.6)], we obtain the series expansion for c:
with all higher-order terms being positive. In particular, \(c(0)=\frac{1}{3}\) and c is strictly increasing. Since \(c(\frac{\pi }{2}) = \frac{4}{\pi ^2}\), the curvature depending term in Lemma 3.1 can be estimated by
whenever \(0<\rho < \frac{\pi }{2\sqrt{b}}\). Analogously for closed balls:
whenever \(0<\rho < \frac{\pi }{2\sqrt{b}}\).
The following lemma is the Riemannian analogue of Corollary 4.5 and Remark 4.6 in [21].
3.3 Lemma
Suppose m, n are positive integers, \(m\le n\), N is an n-dimensional Riemannian manifold, \(p\in N\), \(r=d(p,\cdot )\), \(\rho _0>0\), a is a real-valued Borel function on \(B_{\rho _0}(p)\),
\(V\in {\mathbf {V}}_m(N)\), and \(\Vert \delta V\Vert \) is a Radon measure.
Then, there holds
for all \(0<\sigma<\rho <\rho _0\).
Proof
We proceed similarly as in the proof of Lemma 3.1. Given any \(\sigma< t < \rho \) and any nonnegative non-decreasing smooth function \(\varphi :{\mathbf {R}}\rightarrow {\mathbf {R}}\), we let \(X = \varphi (\frac{r}{t})r\nabla r\) and compute
for all \(T\in {\mathbf {G}}_2(TN)\), where \(\nabla ^Tr\) denotes the orthogonal projection of \(\nabla r\) onto T. Writing
there holds
Adding the term \(m\int _N(1-a(r))\varphi (\frac{r}{t})\,\mathrm d\Vert V\Vert \) on both sides of the inequality, multiplying by \(\frac{1}{t^{m+1}}\), and neglecting positive terms on the right-hand side, we infer
Integrating the inequality with respect to t from \(\sigma \) to \(\rho \), letting \(\varphi \) approach the characteristic function of the interval \((-\infty ,1]\) from above and using Lemma 2.1, it follows
which implies the conclusion. \(\square \)
3.4 Remark
Suppose \(b>0\) and the function a is given by \(a(r) = \sqrt{b}r\cot (\sqrt{b}r)\) for \(0<r<\frac{\pi }{2\sqrt{b}}\). Define the function
Then, using the series expansion of \(\cot (x)\) [see (2.6)] we obtain the series expansion for c:
with all higher-order terms being positive. In particular, \(c(0) = 0\) and c is strictly increasing. Since \(c(\frac{\pi }{2}) = \frac{2}{\pi }\), the curvature depending term in Lemma 3.3 can be estimated by
whenever \(0<t<\frac{\pi }{2\sqrt{b}}\).
The following lemma is a consequence of Lemma 3.3. It can also be derived directly from the first variation formula, see [35, Theorem 5.5].
3.5 Lemma
Suppose m, n are integers, \(2\le m \le n\), N is an n-dimensional Riemannian manifold, \(V \in {\mathbf {V}}_m(N)\), and \(\Vert \delta V\Vert \) is a Radon measure.
Then, there holds \(\Vert V\Vert \{p\} = 0\) for all \(p\in N\).
Proof
Similarly as in (3.3), one can use Fubini’s theorem to compute
where \(r_\sigma = \max \{r,\sigma \}\), and \(r = d(p,\cdot )\). For small \(\rho >0\) we can apply Lemma 2.7 for some \(b>0\) in combination with Lemma 2.1, Lemma 3.3, and (3.6) to infer
Multiplying with \(\sigma ^{m-1}\) implies
Now, only the left-hand side depends on \(\sigma \). Hence, \( {\varvec{\Theta }}^{*1}(\Vert V\Vert ,p)<\infty \) which implies \(\Vert V\Vert \{p\}=0\). \(\square \)
The proof of the following theorem is based on the Euclidean version in the appendix of [18]. See also [32, Corollary 5.8] for the existence of the density.
3.6 Theorem
Suppose n is an integer, \(n\ge 2\), N is an n-dimensional Riemannian manifold, \(V\in {\mathbf {V}}_2(N)\) has generalised mean curvature H, \(H(x) \bot T\) for V almost all \((x,T)\in {\mathbf {G}}_2(N)\), and H is locally square integrable with respect to \(\Vert V\Vert \).
Then, for all \(p\in N{\setminus } {{\,\mathrm{spt}\,}}\Vert V\Vert _{\mathrm {sing}}\), there holds:
-
1.
The density \({\varvec{\Theta }}^2(\Vert V\Vert ,p)\) exists.
-
2.
The function \({\varvec{\Theta }}^2(\Vert V\Vert ,\cdot )\) is upper semi-continuous at p.
Proof
By (3.6) in combination with Lemma 3.5, we have
where \(a_b\) is defined as in Lemma 3.1. We abbreviate
for \(t>0\). Using Hölder’s inequality, we deduce
Moreover, since H is locally square integrable,
Choose \(\rho _0>0\) small enough such that \(B_{\rho _0}(p)\cap {{\,\mathrm{spt}\,}}\Vert \delta V\Vert _{\mathrm {sing}}=\varnothing \) and such that Lemma 3.1 can be applied for some \(b>0\). Then, there holds
for all \(0<\rho <\rho _0\). Hence, putting (3.8), (3.9), and (3.10) into (3.4), we infer
for all \(0<\sigma<\rho <\rho _0\). Applying \(\limsup _{\sigma \rightarrow 0+}\) on the left and \(\liminf _{\rho \rightarrow 0+}\) on the right, it follows
which proves (1). Hence, letting \(\sigma \rightarrow 0\) in (3.4), and using \(2\sqrt{A(t)}\sqrt{W(t)}\le A(t) + W(t)\), we have
for small \(0<\rho <\rho _0\). It follows
On the other hand,
where we used the limit formula for the measure of decreasing sets (see [9, 2.1.3(5)]). Putting (3.11) and (3.12) together and taking the limit \(\rho \rightarrow 0\) implies statement (2). \(\square \)
The following lemma is a generalisation of [5, Eq. (10)].
3.7 Lemma
Suppose n is an integer, \(n\ge 2\), (N, g) is an n-dimensional Riemannian manifold, \(p\in N\), \(V\in {\mathbf {V}}_2(N)\) has generalised mean curvature H, H is square integrable with respect to \(\Vert V\Vert \), \(H(x)\bot T\) for V almost all \((x,T)\in {\mathbf {G}}_2(N)\), \(b<0\), \(\rho _0>0\), the sectional curvature satisfies \(K\le b\) on \({{\,\mathrm{spt}\,}}\Vert V\Vert \cap {{\bar{B}}}_{\rho _0}(p)\), U is a geodesically star-shaped open neighbourhood of p, and \({{\,\mathrm{spt}\,}}\Vert V\Vert \cap {{\bar{B}}}_{\rho _0}(p) \subset U\). Define the functions
\(c_b := s_b'\), and \(\phi :=|b|/(c_b-1)\).
Then, writing \(r=d(p,\cdot )\), there holds
for almost all \(0<\sigma<\rho <\rho _0\).
Proof
Given any nonnegative smooth function \(\varphi :{\mathbf {R}} \rightarrow {\mathbf {R}}\) whose support is contained in the open interval \((-\infty ,\rho _0)\), we define the vector field \(X:=\varphi (r)s_b(r)\nabla r\). Write the metric in polar coordinates \(g = \mathrm dr \otimes \mathrm dr + g_r\) on U and use Lemma 2.5 to estimate
Hence,
which implies
for all \(T\in {\mathbf {G}}_2(TU)\), where \(\nabla ^Tr\) denotes the orthogonal projection of \(\nabla r\) onto T. Writing
and testing the first variation equation [see (1.1)] with X, we infer
Notice that the function \(t\mapsto \Vert V\Vert {{\bar{B}}}_t(p)\) is continuous at \(t_0\) if and only if \(\Vert V\Vert (\{r=t_0\}) = 0\). Since the function \(t\mapsto \Vert V\Vert {{\bar{B}}}_t(p)\) is non-decreasing, it can only have countably many discontinuity points. Choose \(0<\sigma<\rho <\rho _0\) to be continuity points. Define the non-increasing Lipschitz function
and let \(\varphi \) approach \((\phi _\sigma (\cdot ) - \phi (\rho ))_+\), where \((\cdot )_+:=\max \{\cdot ,0\}\). Then, by the dominated convergence theorem, (3.13) becomes
where \(\pi :{\mathbf {G}}_2(N)\rightarrow N\) is the canonical projection. We compute
as well as \(\phi '(r)s_b(r) = -(\phi (r)s_b(r))^2\), and
Putting (3.15) and (3.16) into (3.14) and neglecting negative terms on the right-hand side imply the conclusion. \(\square \)
3.1 Proof of Theorem 1.7
If \(b\ge 0\), the theorem is a consequence of Lemma 3.1 in combination with (3.5). Indeed, by Theorem 3.6, we can multiply the inequality with 4, let \(\sigma \rightarrow 0+\) and let \(\rho \rightarrow \infty \) to conclude the positive curvature case.
Now, suppose that \(b<0\). We are going to determine the limits in Lemma 3.7 as \(\sigma \rightarrow 0+\) and \(\rho \rightarrow \infty \). Using L’Hôspital’s rule twice, one readily verifies
Therefore, by Theorem 3.6,
Similarly, by L’Hôspital’s rule,
Hence, by Hölder’s inequality and square integrability of the generalised mean curvature,
All the other limits can be easily determined using that \({{\,\mathrm{spt}\,}}\Vert V\Vert \) is compact and using that \(p\notin {{\,\mathrm{spt}\,}}\Vert \delta V\Vert _{\mathrm {sing}}\). \(\square \)
4 Diameter bounds
In this Section, we provide the lower diameter bounds for varifolds (Lemmas 4.1 and 4.3) that are needed to prove Theorems 1.2, 1.4 and 1.5. To prove Lemma 4.3, we will need the Hessian comparison theorem of the distance function for asymptotically non-positively curved manifolds (Lemma 4.2). At the end of this section, we will prove Theorem 1.5.
The following lemma is a direct combination of the representation formula for the first variation, Lemma 2.1, with Rauch’s comparison theorem, Lemma 2.7.
4.1 Lemma
Suppose m, n are positive integers, \(m\le n\), N is a complete n-dimensional Riemannian manifold, \(p\in N\), \(b>0\), \(0<\rho <\min \{i_p(N),\frac{\pi }{2\sqrt{b}}\}\), the sectional curvature satisfies \(\sup _{B_\rho (p)}K\le b\), and \(V\in {\mathbf {V}}_m(N)\) satisfies \({{\,\mathrm{spt}\,}}\Vert V\Vert \subset B_\rho (p)\).
Then, there holds
4.2 Lemma
(See [10]) Suppose (N, g) is a complete Riemannian manifold, \(p\in N\), \(r = d(p,\cdot )\), \({{\,\mathrm{Cut}\,}}(p)\) denotes the cut locus of p in N, \(D(p) = N{\setminus }({{\,\mathrm{Cut}\,}}(p)\cup \{p\})\), \(0<b\le 1/4\), and the radial curvature \(K_r\) satisfies
Then, the Hessian \(\nabla ^2r\) of r can be bounded below by
4.3 Lemma
Suppose m, n are positive integers, \(m\le n\), N is a complete n-dimensional Riemannian manifold, \(p\in N\), \(r = d(p,\cdot )\), \({{\,\mathrm{Cut}\,}}(p)\) denotes the cut locus of p in N, \(D(p) = N{\setminus }({{\,\mathrm{Cut}\,}}(p)\cup \{p\})\), \(0<b\le 1/4\), the radial curvature \(K_r\) satisfies \(K_r \le \frac{b}{r^2}\) on D(p), \(\rho >0\), \(V\in {\mathbf {V}}_m(N)\), \({{\,\mathrm{spt}\,}}\Vert V\Vert \subset B_\rho (p)\), and \(\Vert V\Vert ({{\,\mathrm{Cut}\,}}(p))=0\).
Then, there holds
Proof
Using Lemma 4.2, we compute
and thus
for all \(T\in {\mathbf {G}}_m(TD(p))\). Now, we can test the first variation formula (1.1) with the vector field \(r\nabla r\) and use Lemma 2.1 to conclude the proof. \(\square \)
4.1 Proof of Theorem 1.5
The following proof is based on Simon [39, Lemma 1.1].
Let \(0<\rho <\rho _0:=\min \{i,\frac{\pi }{2\sqrt{b}}\}\). Given any \(p\in N\), we notice that \(\Vert V\Vert B_\sigma (p) = \Vert V\Vert {{\bar{B}}}_\sigma (p)\) for all but countably many \(\sigma >0\). Hence, we can combine (1.6) with Theorem 3.6, (3.5), and Lemma 3.1 to deduce
and thus
Define
where \(\frac{1}{0}\) is interpreted \(\infty \) and \(0\cdot \infty \) is interpreted 0. For the rest of the proof, we assume that \(\Vert V\Vert (N)\le C_{(4.2)}(i,b)\). Then, it follows from (4.1) that
which proves (1.7). Now, pick
From (4.2) and (4.3), we deduce \(2\rho \le \min \{i,\frac{\pi }{3\sqrt{b}}\}\). In particular, \(\rho <\rho _0\). If also \(d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )<\min \{i,\frac{\pi }{3\sqrt{b}}\}\), then we can combine (2.8) with Lemma 4.1 and use Hölder’s inequality to infer
which implies \(2\rho \le d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )\) and which proves (1.9). In any case, we are guaranteed that \(2\rho \le d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )\). Now, choose a point \(p\in {{\,\mathrm{spt}\,}}\Vert V\Vert \) such that \(d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert ) = \max _{q\in {{\,\mathrm{spt}\,}}\Vert V\Vert }d(p,q)\) and let \(\nu \) be the integer with
Let \(r:=d(p,\cdot )\) and for each \(j=1,\ldots ,\nu \) choose \(p_j\in {{\,\mathrm{spt}\,}}\Vert V\Vert \cap \{r=\rho j\}\). Then, with \(p_0:=p\), the balls \(B_{\rho /2}(p_0),\ldots ,B_{\rho /2}(p_\nu )\) are pairwise disjoint. Thus, summing over \(j=0,\ldots ,\nu \) in (4.1), we infer
Hence, using \(\nu + 1\ge d_{\mathrm {ext}}({{\,\mathrm{spt}\,}}\Vert V\Vert )/\rho \) and using (4.3), we deduce (1.8) which concludes the proof. \(\square \)
5 Sobolev and isoperimetric inequalities inequalities
In this section, we will prove the Sobolev inequality (see Theorem 1.11) and the isoperimetric inequality (see Corollary 1.12). First, we will need the following lemma.
5.1 Lemma
(See [38, Lemma 18.7] or [35, Lemma 6.3]) Suppose m is a positive integer, f, g are real-valued functions on the interval \((0,\infty )\), f is bounded and non-decreasing,
and
for all \(0<\sigma <\rho \le \rho _0:=(2^{m+1}\lim _{t\rightarrow \infty }f(t))^{1/m}\).
Then, there exists \(0<\rho \le \rho _0\) such that
5.1 Proof of Theorem 1.11
The following proof is an adaptation of [2, Theorem 7.1] and [35, Theorem 6.5].
Let \(h\le 1\) be a nonnegative compactly supported \(C^1\) function on N and define the varifold \(V_h\in {\mathbf {V}}_m(N)\) by letting
for all compactly supported continuous functions k on \({\mathbf {G}}_m(N)\). Given any \(X\in {\mathscr {X}}(N)\), we compute for \(T\in {\mathbf {G}}_m(TN)\) (see [11, Lemma 3.2(ii)])
and thus, by Lemma 2.1,
Given any open set \(U\subset N\), it follows
where \(\pi :{\mathbf {G}}_m(N)\rightarrow N\) is the canonical projection. With the usual approximation from above (see [9, 2.1.3 (5)]), one can see that the inequality remains valid for U replaced with any closed set. In particular, \(\Vert \delta V_h\Vert \) is a Radon measure. Hence, we can apply Lemma 3.3 in combination with Lemmas 2.1 and 2.7, and (3.7) to deduce
for all \(p\in N\) and \(0<\sigma<\rho <\min \{i({{\,\mathrm{spt}\,}}\Vert V\Vert ),\frac{\pi }{2\sqrt{b}}\}\). Define the constant
Given any \(p\in {{\,\mathrm{spt}\,}}\Vert V\Vert \) with \( {\varvec{\Theta }}^{*m}(\Vert V_h\Vert ,p)\ge 1\), Lemma 5.1 applied with \(f(t) = \varvec{\alpha }(m)^{-1}\Vert V_h\Vert {{\bar{B}}}_t(p)\) and
implies by (5.1)
Now, Vitali’s covering theorem (see [9, 2.8.5, 6, 8]) implies the assertion. \(\square \)
5.2 Proof of Corollary 1.12
First, we apply Theorem 1.11 with h approaching the constant function 1 from below, to deduce
Applying this inequality again on the right-hand side, we inductively infer for all positive integers \(k\ge 1\),
provided that \(m\sqrt{b}C_{(5.2)}\Vert V\Vert (N)^{1/m}\le \frac{1}{2}\).
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Acknowledgements
The author is supported by the EPSRC as part of the MASDOC DTC at the University of Warwick, Grant No. EP/HO23364/1. Moreover, I would like thank Andrea Mondino for supervising me, as well as Lucas Ambrozio and Manh Tien Nguyen for discussion on the subject.
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Scharrer, C. Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities. Ann Glob Anal Geom 61, 691–719 (2022). https://doi.org/10.1007/s10455-021-09822-0
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DOI: https://doi.org/10.1007/s10455-021-09822-0
Keywords
- Varifolds on Riemannian manifolds
- Lower diameter bounds for minimal submanifolds
- Li–Yau inequality
- Sobolev inequality
- sectional curvature