Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

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 S+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}_+^3$$\end{document}. 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.


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 in a domain of a Riemannian manifold is called a free boundary submanifold if ∂ ⊂ ∂ and intersects ∂ orthogonally.It is interesting to find sharp area bounds for free boundary minimal surfaces.Fraser-Schoen [6] proved that every free boundary minimal surface in an m-dimensional Euclidean unit ball satisfies Area( ) ≥ π, where equality holds if and only if 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 is an n-dimensional free boundary minimal submanifold in a Euclidean unit ball B m , then where equality holds if and only if is an n-dimensional unit ball B n .Recently,  proved an analogous result for free boundary minimal surfaces in a geodesic ball B m contained in the upper hemisphere S m + as follows: Area( ) ≥ Area(B 2 ), where equality holds if and only if 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 in hyperbolic space H n satisfies the following linear isoperimetric inequality: (n − 1)Vol( ) ≤ Vol(∂ ).
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 in an m-dimensional Euclidean unit ball satisfies the following identity: 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 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 in a Euclidean space satisfies the monotonicity property which states that the volume of ∩ 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).

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., 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 ∈ M, define r (x) to be the distance from p to x in M. For an n-dimensional submanifold in M, the modified volume M k, p ( ) of with center at p is defined as 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 by When M is a space form of constant curvature 0, −1, and 1, we simply denote by M p ( ) the modified volume of .For a submanifold in a manifold M with sectional curvature bounded above by a positive constant, it follows that Moreover, if M has sectional curvature bounded above by a negative constant, we see that Choe-Gulliver [4] proved that if is a minimal surface in the unit upper hemisphere S m + with radially connected boundary ∂ from p, i.e., {s : where equality holds if and only if is a totally geodesic disk centered at p. Now consider an n-dimensional submanifold in an m-dimensional Riemannian manifold M. Fix p ∈ and choose a local orthonormal frame {e 1 , • • • , e n } for in a neighborhood of p.This frame can be smoothly extended to a local orthonormal frame {ē 1 , • • • , ēm } for M in a neighborhood of p, such that ēi = e i for 1 ≤ i ≤ n on .The mean curvature vector H of in M is given by where ∇ and ∇ are the connections of M and , respectively.Thus for f ∈ C ∞ (M) we have where and are the Laplacians on 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 R m , the unit sphere S m , and the hyperbolic space 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): 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 be an n-dimensional submanifold in a complete simply connected Riemannian manifold M m with sectional curvature bounded above by a constant K .Let ∇ and be the connection and Laplacian on , respectively, and ∇ the connection on M. Define r (•) = dist( p, •) for a fixed point p ∈ M. Denote by H the mean curvature vector of .

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 in an m-dimensional Euclidean unit ball satisfies that 2Area( ) = Length(∂ ). ( This result can be extended to free boundary submanifolds in geodesic balls of a space form in terms of the modified volume as follows.Proof.For (i), applying Lemma 2.2 (i), we see that where ν is the outward unit conormal of along ∂ .Since meets ∂ B p (R) orthogonally along ∂ by the free boundary condition, Again, from the free boundary condition, it follows that Thus we get One can prove the assertion (iii) in the same manner by using Lemma 2.2 (iii).
In particular, when 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 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 ∈ M. Denote by H the mean curvature vector of .
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 S 3 + .Corollary 2.5.Let be a free boundary minimal surface in a geodesic ball B p (R) of the upper hemisphere S 3 + with radially connected boundary ∂ from p, i.e., {s : s = dist( p, q), q ∈ ∂ } is a connected interval, then we have where equality holds if and only if is a totally geodesic disk centered at p. Proof.Since is a minimal surface, Corollary 2.4 (iii) implies Using the isoperimetric inequality (1) for the modified area, we have 2π sin R ≤ Length(∂ ).
In other words, Furthermore, we see that equality holds if and only if is a totally geodesic disk centered at p by (1).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 H 3 .It should be mentioned that the corresponding isoperimetric inequality (1) for minimal surfaces in 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 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 ∈ M. Let r (•) be the distance function from p in M. Denote by H the mean curvature vector of .
Proof.By Lemma 2.2 (ii), we have On the other hand, a simple computation shows that and we conclude that which completes the proof of the assertion (ii).
In [4,14,17], it was shown that the following linear isoperimetric inequality holds for any n-dimensional submanifold in hyperbolic space H m , which says that the volume of is bounded above by the volume of ∂ and the integral of H . Theorem 2.7 gives a lower bound of the volume of in terms of Vol(∂ ) and H for a free boundary submanifold 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 in H m M p ( ) ≥ Vol( ) 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 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.r q ≥ 0, where q = n − α R.
Proof.Define s = ∩ B p (s) for 0 < s < R. To see the assertion (i), we use Lemma 2.2 (i).Then we have which gives Apply the coarea formula to deduce where we used the fact that s 1 − |∇r | 2 is nondecreasing in s in the last inequality since 0 ≤ |∇r | ≤ 1.Thus we get which completes the proof of the assertion (i).For (ii), using Lemma 2.
which completes the proof of the assertion (ii).One can prove the assertion (iii) in the same manner.
In particular, if is a minimal submanifold in a geodesic ball B p (R) such that ∂ ⊂ ∂ B p (R), then the monotonicity of the modified volume of 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 be an n-dimensional minimal submanifold in a geodesic ball B Moreover, when the ambient space M is a space form, equality in the above three cases holds if and only if 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 is a cone over the point p, then 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 |∇r | = 1 on .Therefore, it follows that is a cone over a point p with density at p equal to 1.In other words, is an n-dimensional totally geodesic ball centered at p.A similar proof holds for the case where K = 1 or −1.
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Theorem 2 . 3 .
Let 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 ∈ M. Let r (•) be the distance function from p in M. Denote by H the mean curvature vector of .(i) If M = R m , then we have nVol( ) − |H | ≤ Vol(∂ )R ≤ nVol( ) + |H |r.(ii) If M = H m , then we have nM p ( ) − |H | sinh r ≤ Vol(∂ ) sinh R ≤ nM p ( ) + |H | sinh r. (iii) If M = S m and R < π 2 , then we have nM p ( ) − |H | sin r ≤ Vol(∂ ) sin R ≤ nM p ( ) + |H | sin r.

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 be an n-dimensional submanifold in a geodesic ball B p (R) ⊂ M with radius R > 0 centered at p in M such that ∂ ⊂ ∂ B p (R). Define r (•) = dist( p, •).Suppose that the mean curvature vector H of satisfies |H | ≤ α.For 0 < < R, the following holds.