Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

Abstract

We derive linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume. It is known that the twice of the area of a free boundary minimal surface in a Euclidean unit ball is equal to the length of its boundary. This can be extended to space forms by using our linear isoperimetric inequalities for the modified volume. Moreover, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces in a geodesic ball of the upper hemisphere $\mathbb {S}_+^3$. Finally, it is proved that the monotonicity property still holds for the modified volume of any submanifold in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

1 Introduction

Free boundary minimal surfaces have been one of the most intensively investigated topics of differential geometry in recent years. Recall that a properly immersed submanifold $\Sigma$ in a domain $\Omega$ of a Riemannian manifold is called a free boundary submanifold if $\partial \Sigma \subset \partial \Omega$ and $\Sigma$ intersects $\partial \Omega$ orthogonally. It is interesting to find sharp area bounds for free boundary minimal surfaces. Fraser–Schoen [6] proved that every free boundary minimal surface $\Sigma$ in an m-dimensional Euclidean unit ball satisfies

$$\begin{aligned} \mathrm{Area}(\Sigma ) \ge \pi , \end{aligned}$$

where equality holds if and only if $\Sigma$ is a unit disk. Later, Brendle [2] showed that this result is still true for arbitrary free boundary minimal submanifolds in an m-dimensional Euclidean unit ball. In other words, if $\Sigma$ is an n-dimensional free boundary minimal submanifold in a Euclidean unit ball $B^m$, then

$$\begin{aligned} \mathrm{Vol}(\Sigma ) \ge \mathrm{Vol}(B^n), \end{aligned}$$

where equality holds if and only if $\Sigma$ is an n-dimensional unit ball $B^n$. Recently, Freidin–McGrath [8] proved an analogous result for free boundary minimal surfaces $\Sigma$ in a geodesic ball $B^m$ contained in the upper hemisphere ${\mathbb {S}}^m_+$ as follows:

$$\begin{aligned} \mathrm{Area}(\Sigma ) \ge \mathrm{Area}(B^2), \end{aligned}$$

where equality holds if and only if $\Sigma$ is a totally geodesic disk $B^2$ with the same radius as $B^m$ (see also [7]). On the other hand, Yau [17] proved that a domain $\Sigma$ in hyperbolic space ${\mathbb {H}}^n$ satisfies the following linear isoperimetric inequality:

$$\begin{aligned} (n-1) \mathrm{Vol}(\Sigma ) \le \mathrm{Vol}(\partial \Sigma ). \end{aligned}$$

In [4], Choe and Gulliver showed this inequality is still true for minimal submanifolds in hyperbolic space. See also [11] for sharp linear isoperimetric inequalities for minimal submanifolds in hyperbolic space.

In this paper, we investigate linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume (see Definition 2.1), which was introduced by Choe–Gulliver [4]. In Sect. 2, we discuss linear isoperimetric inequalities for the modified volume of a free boundary submanifold in a geodesic ball of space forms (see Theorem 2.3). It is known that a free boundary minimal surface $\Sigma$ in an m-dimensional Euclidean unit ball satisfies the following identity:

$$\begin{aligned} 2\mathrm{Area}(\Sigma ) = \mathrm{Length}(\partial \Sigma ), \end{aligned}$$

which was observed by Fraser–Schoen [6]. This identity can be extended into space forms in Corollary 2.4. Applying our linear isoperimetric inequalities for the modified volume of a free boundary submanifold, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces with radially connected boundary in a geodesic ball of the upper hemisphere ${\mathbb {S}}_+^3$ (see Corollary 2.5). Furthermore, we derive linear isoperimetric inequalities for free boundary submanifolds in a Riemannian manifold with sectional curvature bounded above by a constant (see Theorem 2.10).

In Sect. 3, we study the monotonicity of volume of submanifolds in a Riemannian manifold. It is well-known that an n-dimensional minimal submanifold $\Sigma$ in a Euclidean space satisfies the monotonicity property which states that the volume of $\Sigma \cap B_p (r)$ divided by the volume of n-dimensional ball of radius r is a nondecreasing function of r (see [5, 16] for instance). The monotonicity of minimal submanifolds still holds in hyperbolic space, but not in the unit sphere (see [1, 9]). Choe–Gulliver [4] proved the monotonicity of minimal submanifolds in space forms in terms of the modified volume. Later, the second author [13, 15] showed that minimal submanifolds outside a convex set in space forms enjoy the monotonicity of the modified volume in a relative sense. It is proved that the monotonicity property still holds for the modified volume of (not necessarily minimal) submanifolds in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant (see Theorem 3.1).

2 Linear isoperimetric inequalities

We first define the modified volume of a submanifold in a Riemannian manifold with sectional curvature bounded above by a constant as follows.

Definition 2.1

(Choe–Gulliver [4], Seo [15]). Let M be a complete simply connected Riemannian manifold with sectional curvature bounded above by a positive constant $K=k^2$ for $k>0$. Given $p \in M$, define r(x) to be the distance from p to x in M. For an n-dimensional submanifold $\Sigma$ in M, the modified volume $M_{k, p} (\Sigma )$ of $\Sigma$ with center at p is defined as

$$\begin{aligned} M_{k, p} (\Sigma ) = \int _\Sigma \cos kr . \end{aligned}$$

Similarly, when M is a complete simply connected Riemannian manifold with sectional curvature bounded above by a negative constant $K=-k^2$ for $k>0$, we define the modified volume of $\Sigma$ by

$$\begin{aligned} M_{k, p} (\Sigma ) = \int _\Sigma \cosh kr . \end{aligned}$$

When M is a space form of constant curvature $0, -1,$ and 1, we simply denote by $M_p (\Sigma )$ the modified volume of $\Sigma$. For a submanifold $\Sigma$ in a manifold M with sectional curvature bounded above by a positive constant, it follows that

$$\begin{aligned} M_{k, p} (\Sigma ) \le \mathrm{Vol}(\Sigma ). \end{aligned}$$

Moreover, if M has sectional curvature bounded above by a negative constant, we see that

$$\begin{aligned} M_{k, p} (\Sigma ) \ge \mathrm{Vol}(\Sigma ). \end{aligned}$$

Choe–Gulliver [4] proved that if $\Sigma$ is a minimal surface in the unit upper hemisphere ${\mathbb {S}}^m_+$ with radially connected boundary $\partial \Sigma$ from p, i.e., $\{s: s=\mathrm{dist}(p, q), q \in \partial \Sigma \}$ is a connected interval, then

$$\begin{aligned} 4\pi M_p (\Sigma ) \le \mathrm{Length}(\partial \Sigma )^2, \end{aligned}$$

(1)

where equality holds if and only if $\Sigma$ is a totally geodesic disk centered at p.

Now consider an n-dimensional submanifold $\Sigma$ in an m-dimensional Riemannian manifold M. Fix $p \in \Sigma$ and choose a local orthonormal frame $\{e_1, \cdots , e_n\}$ for $\Sigma$ in a neighborhood of p. This frame can be smoothly extended to a local orthonormal frame $\{{\bar{e}}_1, \cdots , {\bar{e}}_m\}$ for M in a neighborhood of p, such that ${\bar{e}}_i = e_i$ for $1\le i \le n$ on $\Sigma$. The mean curvature vector H of $\Sigma$ in M is given by

$$\begin{aligned} H = \sum _{i=1}^n {\overline{\nabla }}_{{\bar{e}}_i} {\bar{e}}_i - \sum _{i=1}^n \nabla _{e_i} e_i, \end{aligned}$$

where ${\overline{\nabla }}$ and $\nabla$ are the connections of M and $\Sigma$, respectively. Thus for $f\in C^{\infty }(M)$ we have

$$\begin{aligned} \Delta f&= \sum _{i=1}^n (e_i e_i f - \nabla _{e_i}e_i f)\\&= \sum _{i=1}^m {\bar{e}}_i {\bar{e}}_i f - \sum _{\alpha =n+1}^m {\bar{e}}_\alpha {\bar{e}}_\alpha f - \sum _{i=1}^m {\overline{\nabla }}_{{\bar{e}}_i} {\bar{e}}_i f + Hf + \sum _{\alpha =n+1}^m {\overline{\nabla }}_{{\bar{e}}_\alpha } {\bar{e}}_\alpha f \\&= {\overline{\Delta }} f + Hf - \sum _{\alpha =n+1}^m {\overline{\nabla }}^2 f ({\bar{e}}_\alpha \, , {\bar{e}}_\alpha ) , \end{aligned}$$

where $\Delta$ and ${\overline{\Delta }}$ are the Laplacians on $\Sigma$ and M, respectively. In particular, if $M^m$ is an m-dimensional space form, i.e., an m-dimensional complete simply-connected Riemannian manifold with constant sectional curvature K, then, up to homotheties, we may assume that $K=0, 1$, and $-1$: the corresponding spaces are the Euclidean space ${\mathbb {R}}^m$, the unit sphere ${\mathbb {S}}^m$, and the hyperbolic space ${\mathbb {H}}^m$, respectively. For the distance r(x) from a fixed point p to x on M, the following fact is well-known (see [12] for example):

$$\begin{aligned} \left\{ \begin{array}{llll} \mathrm{Hess}_M \frac{1}{2}r^2&{}=&{} g &{}~~\text {if } M ={\mathbb {R}}^m,\\ \mathrm{Hess}_M \cosh r&{}=&{} \cosh r \cdot g &{}~~\text {if } M ={\mathbb {H}}^m,\\ \mathrm{Hess}_M \cos r&{}=&{} -\cos r \cdot g &{}~~\text {if } M= {\mathbb {S}}^m ~~\text {and}~~ r< \frac{\pi }{2}, \end{array} \right. \end{aligned}$$

where g denotes the metric on M. Combining these observations together, we get the following lemma, which was proved by Choe–Gulliver [4] in the case when $H=0$ (see also [3, 10, 14]).

Lemma 2.2

([3, 4, 10, 14]). Let $\Sigma$ be an n-dimensional submanifold in a complete simply connected Riemannian manifold $M^m$ with sectional curvature bounded above by a constant K. Let $\nabla$ and $\Delta$ be the connection and Laplacian on $\Sigma$, respectively, and ${\overline{\nabla }}$ the connection on M. Define $r(\cdot ) = \mathrm{dist}(p, \cdot )$ for a fixed point $p\in M$. Denote by H the mean curvature vector of $\Sigma$.

• (i) If $K=0$, then we have

  (a) $\Delta r^2 \ge 2n + 2r \langle H, {\overline{\nabla }} r\rangle$,

  (b) $\Delta r \ge \frac{n-|\nabla r|^2}{r} + \langle H, {\overline{\nabla }} r\rangle$.

• (ii) If $K=-k^2$ for $k>0$, then

  (c) $\Delta r \ge k(n-|\nabla r|^2) \coth kr + \langle H, {\overline{\nabla }} r\rangle$,

  (d) $\Delta \cosh kr \ge nk^2\cosh kr + k\sinh kr \langle H, {\overline{\nabla }} r \rangle$.

• (iii) If $K=k^2$ for $k>0$ and $r < \frac{\pi }{2k}$, then

  (e) $\Delta r \ge k(n-|\nabla r|^2) \cot kr + \langle H, {\overline{\nabla }} r\rangle$,

  (f) $\Delta \cos kr \le -nk^2\cos kr - k\sin kr \langle H, {\overline{\nabla }} r \rangle$.

Moreover, equality holds in the above inequalities if M has constant sectional curvature.

In 2011, Fraser–Schoen [6] observed that a free boundary minimal surface $\Sigma$ in an m-dimensional Euclidean unit ball satisfies that

$$\begin{aligned} 2\mathrm{Area}(\Sigma ) = \mathrm{Length}(\partial \Sigma ). \end{aligned}$$

(2)

This result can be extended to free boundary submanifolds in geodesic balls of a space form in terms of the modified volume as follows.

Theorem 2.3

Let $\Sigma$ be an n-dimensional free boundary submanifold in $B_{p} (R)$ of a space form $M^m$, where $B_{p} (R)$ denotes a geodesic ball of radius R centered at $p\in M$. Let $r(\cdot )$ be the distance function from p in M. Denote by H the mean curvature vector of $\Sigma$.

• (i) If $M= {\mathbb {R}}^m$, then we have

  $$\begin{aligned}&n \mathrm{Vol}(\Sigma ) - \int _{\Sigma } |H| \le \mathrm{Vol}(\partial \Sigma ) R \le n \mathrm{Vol}(\Sigma ) \\&\quad + \int _{\Sigma } |H| r. \end{aligned}$$

• (ii) If $M= {\mathbb {H}}^m$, then we have

  $$\begin{aligned}&n M_p(\Sigma ) - \int _{\Sigma } |H|\sinh r \le \mathrm{Vol}(\partial \Sigma ) \sinh R \le n M_p(\Sigma ) \\&\quad + \int _{\Sigma } |H| \sinh r. \end{aligned}$$

• (iii) If $M= {\mathbb {S}}^m$ and $R < \frac{\pi }{2}$, then we have

  $$\begin{aligned} n M_p(\Sigma )-\int _{\Sigma } |H| \sin r \le \mathrm{Vol}(\partial \Sigma ) \sin R \le n M_p(\Sigma )+ \int _{\Sigma } |H| \sin r. \end{aligned}$$

Proof

For (i), applying Lemma 2.2 (i), we see that

$$\begin{aligned} n-r|H| \le \frac{1}{2} \Delta r^2 \le n +r|H|, \end{aligned}$$

which implies

$$\begin{aligned} n \mathrm{Vol}(\Sigma ) - \int _{\Sigma } r |H| \le \int _{\partial \Sigma } r \frac{\partial r}{\partial \nu } \le n \mathrm{Vol}(\Sigma ) + \int _{\Sigma } r|H| \end{aligned}$$

where $\nu$ is the outward unit conormal of $\Sigma$ along $\partial \Sigma$. Since $\Sigma$ meets $\partial B_p(R)$ orthogonally along $\partial \Sigma$ by the free boundary condition,

$$\begin{aligned} n \mathrm{Vol}(\Sigma ) - \int _{\Sigma } r |H| \le \mathrm{Vol}(\partial \Sigma ) R \le n \mathrm{Vol}(\Sigma ) + \int _{\Sigma } r |H|. \end{aligned}$$

For (ii), Lemma 2.2 (ii) gives

$$\begin{aligned} \int _{\partial \Sigma } \sinh r \,\frac{\partial r}{\partial \nu } - n \,M_p(\Sigma ) =\int _{\Sigma } \sinh r \left\langle H, {\overline{\nabla }} r \right\rangle . \end{aligned}$$

Again, from the free boundary condition, it follows that

$$\begin{aligned} \mathrm{Vol}(\partial \Sigma )\sinh R - n M_p(\Sigma ) = \int _{\Sigma }\sinh r \left\langle H, {\overline{\nabla }} r \right\rangle . \end{aligned}$$

Thus we get

$$\begin{aligned} n M_p(\Sigma ) - \int _{\Sigma } |H|\sinh r \le \mathrm{Vol}(\partial \Sigma ) \sinh R \le n M_p(\Sigma ) + \int _{\Sigma } |H| \sinh r. \end{aligned}$$

One can prove the assertion (iii) in the same manner by using Lemma 2.2 (iii). $\square$

In particular, when $\Sigma$ is a free boundary minimal submanifold in a geodesic ball, we have the following, which can be regarded as an extension of (2) due to Fraser–Schoen.

Corollary 2.4

Let $\Sigma$ be an n-dimensional free boundary minimal submanifold in $B_{p} (R)$ of a space form $M^m$, where $B_{p} (R)$ denotes a geodesic ball of radius R centered at $p\in M$. Denote by H the mean curvature vector of $\Sigma$.

• (i) If $M= {\mathbb {R}}^m$, then

  $$\begin{aligned} n \mathrm{Vol}(\Sigma ) = \mathrm{Vol}(\partial \Sigma ) R . \end{aligned}$$

• (ii) If $M= {\mathbb {H}}^m$, then

  $$\begin{aligned} n M_p(\Sigma ) = \mathrm{Vol}(\partial \Sigma ) \sinh R . \end{aligned}$$

• (iii) If $M= {\mathbb {S}}^m$ and $R < \frac{\pi }{2}$, then

  $$\begin{aligned} n M_p(\Sigma )= \mathrm{Vol}(\partial \Sigma ) \sin R . \end{aligned}$$

As another application of Theorem 2.3, we are able to obtain a sharp lower bound for the modified volume of free boundary minimal surfaces with radially connected boundary in a geodesic ball of the upper hemisphere ${\mathbb {S}}_+^3$ .

Corollary 2.5

Let $\Sigma$ be a free boundary minimal surface in a geodesic ball $B_{p} (R)$ of the upper hemisphere ${\mathbb {S}}_+^3$ with radially connected boundary $\partial \Sigma$ from p, i.e., $\{s: s=\mathrm{dist}(p, q), q \in \partial \Sigma \}$ is a connected interval, then we have

$$\begin{aligned} \pi \sin ^2 R \le M_p(\Sigma ), \end{aligned}$$

where equality holds if and only if $\Sigma$ is a totally geodesic disk centered at p.

Proof

Since $\Sigma$ is a minimal surface, Corollary 2.4 (iii) implies

$$\begin{aligned} 2 M_p(\Sigma )= \mathrm{Length}(\partial \Sigma ) \sin R. \end{aligned}$$

Using the isoperimetric inequality (1) for the modified area, we have

$$\begin{aligned} 2\pi \sin R \le \mathrm{Length}(\partial \Sigma ). \end{aligned}$$

In other words,

$$\begin{aligned} \pi \sin ^2 R \le M_p(\Sigma ). \end{aligned}$$

Furthermore, we see that equality holds if and only if $\Sigma$ is a totally geodesic disk centered at p by (1). $\square$

Remark 2.6

It would be interesting to remove the assumption that the boundary is radially connected from p in Corollary 2.5. Moreover, it seems plausible that an analogue of Corollary 2.5 holds for free boundary minimal surfaces in ${\mathbb {H}}^3$. It should be mentioned that the corresponding isoperimetric inequality (1) for minimal surfaces in ${\mathbb {H}}^3$ is no longer valid.

For free boundary submanifolds in a geodesic ball of a space form, we have the following, which is different from Theorem 2.3.

Theorem 2.7

Let $\Sigma$ be an n-dimensional free boundary submanifold in $B_{p} (R)$ of a space form $M^m$, where $B_{p} (R)$ denotes a geodesic ball of radius R centered at $p\in M$. Let $r(\cdot )$ be the distance function from p in M. Denote by H the mean curvature vector of $\Sigma$.

(i) If $M={\mathbb {H}}^m$, then we have

$$\begin{aligned}&\frac{n}{2} \mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{\sinh r}{1+ \cosh r} |H| \le \frac{\sinh R}{1+\cosh R} \mathrm{Vol}(\partial \Sigma ) \le \frac{n \cosh R}{1+\cosh R} \mathrm{Vol}(\Sigma ) \\&\quad + \int _{\Sigma } \frac{\sinh r}{1+\cosh r} |H|. \end{aligned}$$

(ii) If $M={\mathbb {S}}^m$ and $R <\frac{\pi }{2}$, then

$$\begin{aligned}&\frac{n \cos R}{1+ \cos R} \mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{\sin r}{1+\cos r} |H| \le \frac{\sin R}{1+\cos R} \mathrm{Vol}(\partial \Sigma ) \le \frac{n}{2} \mathrm{Vol}(\Sigma ) \\&\quad + \int _{\Sigma } \frac{\sin r}{1+\cos r} |H|. \end{aligned}$$

Proof

By Lemma 2.2 (ii), we have

$$\begin{aligned} \Delta r = (n-|\nabla r|^2) \coth r + \langle H, {\overline{\nabla }} r \rangle . \end{aligned}$$

It follows that

$$\begin{aligned} \Delta \log (1 + \cosh r)&=\text {div}\left( \frac{\sinh r}{1+\cosh r} \nabla r \right) \\&=\frac{\cosh r(1+\cosh r)-\sinh ^2 r}{(1+\cosh r)^2} |\nabla r|^2 +\frac{\sinh r \Delta r}{1+\cosh r}\\&=\frac{1}{1+\cosh r} |\nabla r|^2 + \frac{\sinh r}{1+\cosh r}\left( (n-|\nabla r|^2) \coth r + \langle H, {\overline{\nabla }} r\rangle \right) \\&=\frac{1}{1+\cosh r} \left( n\cosh r + (1-\cosh r) |\nabla r|^2 + \langle H, {\overline{\nabla }} r \rangle \sinh r \right) , \end{aligned}$$

which gives

$$\begin{aligned} \frac{(n-1)\cosh r +1 -|H| \sinh r}{1+\cosh r} \le \Delta \log (1+ \cosh r) \le \frac{n\cosh r +|H| \sinh r}{1+\cosh r}. \end{aligned}$$

On the other hand, a simple computation shows that

$$\begin{aligned} \dfrac{(n-1) \cosh r +1}{1+\cosh r} \ge \dfrac{n}{2}. \end{aligned}$$

Thus we see that

$$\begin{aligned} \int _{\Sigma } \frac{(n-1)\cosh r +1 -|H| \sinh r}{1+\cosh r} \ge \frac{n}{2} \mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{|H|\sinh r}{1+\cosh r} \end{aligned}$$

and

$$\begin{aligned} \int _{\Sigma } \frac{n\cosh r +|H| \sinh r}{1+\cosh r} \le \frac{n \cosh R}{1+\cosh R} \mathrm{Vol}(\Sigma ) + \int _{\Sigma } \frac{|H|\sinh r}{1+\cosh r}. \end{aligned}$$

Applying the free boundary condition, we get

$$\begin{aligned} \int _{\Sigma } \Delta \log (1+ \cosh r)&= \int _{\Sigma } \text {div} \, \left( \frac{\sinh r}{1+\cosh r} \nabla r \right) \\&= \int _{\partial \Sigma } \frac{\sinh r}{1+\cosh r} \frac{\partial r}{\partial \nu } \\&= \frac{\sinh R}{1+\cosh R} \mathrm{Vol}(\partial \Sigma ). \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} \frac{n}{2}\mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{\sinh r}{1+ \cosh r} |H|&\le \frac{\sinh R}{1+\cosh R} \mathrm{Vol}(\partial \Sigma ) \\&\le \frac{n\cosh R}{1+\cosh R}\mathrm{Vol}(\Sigma ) + \int _{\Sigma } \frac{\sinh r}{1+\cosh r} |H|, \end{aligned}$$

which completes the proof of the assertion (i).

In order to prove the assertion (ii), by Lemma 2.2 (iii), we have

$$\begin{aligned} \Delta r = (n-|\nabla r|^2) \cot r + \langle H, {\overline{\nabla }} r \rangle . \end{aligned}$$

We compute

$$\begin{aligned} -\Delta \log (1 + \cos r)&=\text {div}\left( \frac{\sin r}{1+\cos r} \nabla r \right) \\&=\frac{\cos r(1+\cos r)+\sin ^2 r}{(1+\cos r)^2} |\nabla r|^2 +\frac{\sin r \Delta r}{1+\cos r}\\&=\frac{1}{1+\cos r} |\nabla r|^2 + \frac{\sin r}{1+\cos r}\left( (n-|\nabla r|^2) \cot r + \langle H, {\overline{\nabla }} r\rangle \right) \\&=\frac{1}{1+\cos r} \left( n\cos r + (1-\cos r) |\nabla r|^2 + \langle H, {\overline{\nabla }} r \rangle \sin r \right) . \end{aligned}$$

Thus

$$\begin{aligned} \frac{n\cos r -|H| \sin r}{1+\cos r} \le -\Delta \log (1+ \cosh r) \le \frac{(n-1)\cos r +1 +|H| \sin r}{1+\cos r}. \end{aligned}$$

Since

$$\begin{aligned} \int _{\Sigma } -\Delta \log (1+ \cos r)&= \int _{\Sigma } \text {div} \left( \frac{\sin r}{1+\cos r} \nabla r \right) \\&= \int _{\partial \Sigma } \frac{\sin r}{1+\cos r} \frac{\partial r}{\partial \nu } \\&= \frac{\sin R}{1+\cos R} \mathrm{Vol}(\partial \Sigma ) \end{aligned}$$

and

$$\begin{aligned} \dfrac{(n-1) \cos r +1}{1+\cos r} \le \dfrac{n}{2}, \end{aligned}$$

we conclude that

$$\begin{aligned}&\frac{n\cos R}{1+\cos R}\mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{\sin r}{1+\cos r} |H| \le \frac{\sin R}{1+\cos R} \mathrm{Vol}(\partial \Sigma ) \le \frac{n}{2}\mathrm{Vol}(\Sigma ) \\&\quad + \int _{\Sigma } \, \frac{\sin r}{1+ \cos r} |H|, \end{aligned}$$

which completes the proof of the assertion (ii). $\square$

In [4, 14, 17], it was shown that the following linear isoperimetric inequality

$$\begin{aligned} (n-1) \mathrm{Vol}(\Sigma ) \le \mathrm{Vol}(\partial \Sigma ) + \int _\Sigma |H| \end{aligned}$$

holds for any n-dimensional submanifold $\Sigma$ in hyperbolic space ${\mathbb {H}}^m$, which says that the volume of $\Sigma$ is bounded above by the volume of $\partial \Sigma$ and the integral of H. Theorem 2.7 gives a lower bound of the volume of $\Sigma$ in terms of $\mathrm{Vol}(\partial \Sigma )$ and H for a free boundary submanifold $\Sigma$ in a geodesic ball of a space form, which can be regarded as an extension of [4, 14, 17].

Remark 2.8

Theorem 2.7 provides upper and lower bounds on the boundary volume of a free boundary submanifold in a geodesic ball of a space form in terms of the usual interior volume, whereas Theorem 2.3 provides the bounds in terms of the modified volume. By the definition of the modified volume, it is easy to see that for a submanifold $\Sigma$ in ${\mathbb {H}}^m$

$$\begin{aligned} M_p (\Sigma ) \ge \mathrm{Vol}(\Sigma ) \end{aligned}$$

and for $\Sigma \subset {\mathbb {S}}^m$

$$\begin{aligned} M_p (\Sigma ) \le \mathrm{Vol}(\Sigma ). \end{aligned}$$

Moreover, equality holds in Theorem 2.3 when the free boundary submanifold is minimal. However, the inequalities in Theorem 2.7 are not sharp even for free boundary minimal submanifolds. This observation shows that Theorem 2.3 and Theorem 2.7 are not directly comparable.

As a consequence of Theorem 2.7, when $\Sigma$ is a free boundary minimal surface in a geodesic ball of a space form, we can deduce the following isoperimetric inequalities from Theorem 2.7.

Corollary 2.9

Let $\Sigma$ be a free boundary minimal surface in $B_{p} (R)$ of a space form $M^m$, where $B_{p} (R)$ denotes a geodesic ball of radius R centered at $p\in M$.

• (i) If $M={\mathbb {H}}^3$, then we have

  $$\begin{aligned} \mathrm{Area}\, (\Sigma ) \le \dfrac{\sinh R}{1+\cosh R} \, \mathrm{Length}\, (\partial \Sigma ) \le \frac{2\cosh R}{1+\cosh R} \mathrm{Area}\,(\Sigma ). \end{aligned}$$

• (ii) If $M={\mathbb {S}}^3$ and $R <\frac{\pi }{2}$, then

  $$\begin{aligned} \frac{2 \cos R}{ 1+\cos R} \, \mathrm{Area}\,(\Sigma ) \le \frac{\sin R}{1+\cos R} \, \mathrm{Length}\, (\partial \Sigma ) \le \mathrm{Area}\,(\Sigma ). \end{aligned}$$

So far we have obtained linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a space form. It turns out that such linear isoperimetric inequalities are still valid in a complete simply connected Riemannian manifold $M^m$ with sectional curvature bounded above by a constant. By using the same argument as in the proof of Theorem 2.7, we are able to prove linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold with varying sectional curvature as follows.

Theorem 2.10

Let $M^m$ be an m-dimensional complete simply connected Riemannian manifold $M^m$ with sectional curvature bounded above by a constant K. Let $\Sigma$ be an n-dimensional free boundary submanifold in a geodesic ball $B_{p} (R) \subset M$ with radius $R>0$ centered at p in M. Define $r(\cdot ) = \mathrm{dist}(p, \cdot )$ for fixed $p\in M$. Denote by H the mean curvature vector of $\Sigma$.

• (i) If $K =0$, then we have

  $$\begin{aligned} n \mathrm{Vol}(\Sigma ) - \int _{\Sigma } r |H| \le R \, \mathrm{Vol}\,(\partial \Sigma ) \end{aligned}$$

• (ii) If $K= -k^2$ for some $k>0$, then

  $$\begin{aligned} \frac{k n}{2} \mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{\sinh k r}{1+\cosh k r}\,|H| \le \frac{\sinh k R}{1+ \cosh k R} \mathrm{Vol}(\partial \Sigma ) . \end{aligned}$$

• (iii) If $K= k^2$ for some $k>0$ and $R < \dfrac{\pi }{2k}$, then

  $$\begin{aligned} \frac{k n \cos kR}{1+\cos kR} \mathrm{Vol}(\Sigma ) - \int _{\Sigma } \frac{\sin kr}{1+\cos kr} |H| \le \frac{\sin k R}{1+ \cos k R} \mathrm{Vol}(\partial \Sigma ) . \end{aligned}$$

3 Monotonicity

In this section, we prove that the monotonicity property still holds for the modified volume of (not necessarily minimal) submanifolds in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

Theorem 3.1

Let $M^m$ be an m-dimensional complete simply connected Riemannian manifold $M^m$ with sectional curvature bounded above by a constant K. Let $\Sigma$ be an n-dimensional submanifold in a geodesic ball $B_{p} (R) \subset M$ with radius $R>0$ centered at p in M such that $\partial \Sigma \subset \partial B_p (R)$. Define $r(\cdot ) = \mathrm{dist}(p, \cdot )$. Suppose that the mean curvature vector H of $\Sigma$ satisfies $|H| \le \alpha$. For $0<r<R$, the following holds.

• (i) If $K= 0$ and $\alpha R <n$, then

  $$\begin{aligned} \frac{d}{dr}\left( \dfrac{\mathrm{Vol}(\Sigma \cap B_p (r))}{r^q}\right) \ge 0, \text { where } q=n-\alpha R. \end{aligned}$$

• (ii) If $K= -k^2$ for some $k>0$ and $\alpha \tanh kR < nk$, then

  $$\begin{aligned} \frac{d}{dr} \left( \dfrac{M_{k,p}(\Sigma \cap B_p (r))}{\sinh ^ q kr}\right) \ge 0, \text { where } q=n-\frac{\alpha }{k}\tanh kR. \end{aligned}$$

• (iii) If $K= k^2$ for some $k>0$, $R < \dfrac{\pi }{2k}$ and $\alpha \tan kR < nk$, then

  $$\begin{aligned} \frac{d}{dr} \left( \dfrac{M_{k,p}(\Sigma \cap B_p (r))}{\sin ^ q kr}\right) \ge 0, \text { where } q=n-\frac{\alpha }{k}\tan kR. \end{aligned}$$

Proof

Define $\Sigma _s = \Sigma \cap B_p (s)$ for $0<s<R$. To see the assertion (i), we use Lemma 2.2 (i). Then we have

$$\begin{aligned} \frac{1}{2}\Delta r^2 \ge n +r \left\langle H, {\overline{\nabla }} r\right\rangle \ge n- \alpha R, \end{aligned}$$

which gives

$$\begin{aligned} (n-\alpha R) \,\mathrm{Vol}(\Sigma _s) \le \int _{\Sigma _s} \frac{1}{2}\Delta r^2 = \int _{\partial \Sigma _s} r \frac{\partial r}{\partial \nu } \le s \int _{\partial \Sigma _s} |\nabla r|. \end{aligned}$$

Apply the coarea formula to deduce

$$\begin{aligned} (n-\alpha R)\mathrm{Vol}(\Sigma _s)&\le s\frac{d}{ds} \int _{\Sigma _s} |\nabla r|^2 \nonumber \\&= s \frac{d}{ds} \mathrm{Vol}(\Sigma _s) - s \frac{d}{ds} \int _{\Sigma _s} \left( 1-|\nabla r|^2\right) \nonumber \\&\le s \frac{d}{ds} \mathrm{Vol}(\Sigma _s), \end{aligned}$$

(3)

where we used the fact that $\int _{\Sigma _s} \left( 1-|\nabla r|^2\right)$ is nondecreasing in s in the last inequality since $0 \le |\nabla r| \le 1$. Thus we get

$$\begin{aligned} \frac{d}{ds} \left( \frac{\mathrm{Vol}(\Sigma _s)}{s^{n-\alpha R}}\right) \ge 0, \end{aligned}$$

which completes the proof of the assertion (i). For (ii), using Lemma 2.2 (ii), we have

$$\begin{aligned} \Delta \cosh kr \ge nk^2 \cosh kr + k \sinh kr \left\langle H, {\overline{\nabla }}r\right\rangle , \end{aligned}$$

which implies

$$\begin{aligned} nk^2 \int _{\Sigma _s} \cosh kr \le \int _{\partial \Sigma _s} k \sinh kr \frac{\partial r}{\partial \nu } - \int _{\Sigma _s}k \sinh kr \left\langle H, {\overline{\nabla }}r\right\rangle . \end{aligned}$$

Thus

$$\begin{aligned} nk \int _{\Sigma _s} \cosh kr&\le \int _{\partial \Sigma _s} \sinh kr \,|\nabla r| + \alpha \int _{\Sigma _s} \sinh kr \\&= \frac{\sinh ks}{\cosh ks} \int _{\partial \Sigma _s} \cosh kr \, |\nabla r| + \alpha \int _{\Sigma _s} \frac{\sinh kr}{\cosh kr} \cosh kr \\&\le \frac{\sinh ks}{\cosh ks} \frac{d}{ds} \int _{\Sigma _s} \cosh kr \,|\nabla r|^2 + \alpha \, \frac{\sinh kR}{\cosh kR} \int _{\Sigma _s} \cosh kr \end{aligned}$$

by the coarea formula. Since

$$\begin{aligned} \frac{d}{ds} \int _{\Sigma _s} \cosh kr \,|\nabla r|^2&= \frac{d}{ds} \int _{\Sigma _s} \cosh kr - \frac{d}{ds} \int _{\Sigma _s} \cosh kr\left( 1-|\nabla r|^2\right) \\&\le \frac{d}{ds} \int _{\Sigma _s} \cosh kr, \end{aligned}$$

we obtain

$$\begin{aligned}&\,nk \int _{\Sigma _s} \cosh kr \le \, \frac{\sinh ks}{\cosh ks} \frac{d}{ds} \int _{\Sigma _s} \cosh kr+ \alpha \, \tanh kR \int _{\Sigma _s} \cosh kr, \end{aligned}$$

which shows that

$$\begin{aligned} \frac{d}{ds} \log \left( \dfrac{M_{k,p}(\Sigma _s)}{\sinh ^ q ks}\right) \ge 0, \end{aligned}$$

where $q=n-\frac{\alpha }{k}\tanh kR$. This implies that

$$\begin{aligned} \frac{d}{ds} \left( \dfrac{M_{k,p}(\Sigma _s)}{\sinh ^ q ks}\right) \ge 0, \end{aligned}$$

which completes the proof of the assertion (ii). One can prove the assertion (iii) in the same manner. $\square$

In particular, if $\Sigma$ is a minimal submanifold in a geodesic ball $B_p (R)$ such that $\partial \Sigma \subset \partial B_p (R)$, then the monotonicity of the modified volume of $\Sigma$ is given as follows.

Corollary 3.2

Let $M^m$ be an m-dimensional complete simply connected Riemannian manifold $M^m$ with sectional curvature bounded above by a constant K. Let $\Sigma$ be an n-dimensional minimal submanifold in a geodesic ball $B_{p} (R) \subset M$ with radius $R>0$ centered at p in M such that $\partial \Sigma \subset \partial B_p (R)$. Define $r(\cdot ) = \mathrm{dist}(p, \cdot )$. For $0<r<R$, the following holds.

• (i) If $K= 0$, then

  $$\begin{aligned} \frac{d}{dr}\left( \dfrac{\mathrm{Vol}(\Sigma \cap B_p (r))}{r^n}\right) \ge 0. \end{aligned}$$

• (ii) If $K= -k^2$ for some $k>0$, then

  $$\begin{aligned} \frac{d}{dr} \left( \dfrac{M_{k,p}(\Sigma \cap B_p (r))}{\sinh ^n kr}\right) \ge 0. \end{aligned}$$

• (iii) If $K= k^2$ for some $k>0$ and $R < \dfrac{\pi }{2k}$, then

  $$\begin{aligned} \frac{d}{dr} \left( \dfrac{M_{k,p}(\Sigma \cap B_p (r))}{\sin ^n kr}\right) \ge 0. \end{aligned}$$

Moreover, when the ambient space M is a space form, equality in the above three cases holds if and only if $\Sigma$ is an n-dimensional totally geodesic ball centered at p.

Proof

The assertions (i), (ii), and (iii) immediately follow from Theorem 3.1. Now consider the equality case when the ambient space is a space form. Obviously, if $\Sigma$ is a cone over the point p, then $\Sigma$ satisfies equality in three cases. Conversely, suppose equality holds the above three cases. Observe that every inequality in Lemma 2.2 becomes equality when M is a space form. Furthermore, in case where $K=0$, we have equality in (3), which shows that $|\nabla r| = 1$ on $\Sigma$. Therefore, it follows that $\Sigma$ is a cone over a point p with density at p equal to 1. In other words, $\Sigma$ is an n-dimensional totally geodesic ball centered at p. A similar proof holds for the case where $K=1$ or $-1$. $\square$