Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold

Abstract

Let be a domain on an -dimensional minimal submanifold in the outside of a convex set in or . The modified volume is introduced by Choe and Gulliver (1992) and we prove a sharp modified relative isoperimetric inequality for the domain , , where is the volume of the unit ball of . For any domain on a minimal surface in the outside convex set in an -dimensional Riemannian manifold, we prove a weak relative isoperimetric inequality , where is an upper bound of sectional curvature of the Riemannian manifold.

1. Introduction

Let be the simple closed curve of a domain in a two-dimensional space form with constant curvature . Then the well-known sharp isoperimetric inequality is the following:

(1.1)

where equality holds if and only if is a geodesic disk (see [1]).

An immediate consequence of this inequality is that if is a closed half-space of a two-dimensional space form with constant curvature and is a domain in with , then

(1.2)

where equality holds if and only if is a totally geodesic half-disk with the geodesic part of its boundary contained in . This follows the original isoperimetric inequality after extending the domain by mirror symmetry with respect to .

Motivated by this, one arises natural questions as follows.

If is a convex set in an -dimensional space form with constant curvature and is a minimal surface in the outside of with , does satisfy inequality

(1.3)

How about an -dimensional minimal submanifold case?

Equation (1.3) is called the relative isoperimetric inequality, is called the supporting set of , and is called the relative area of . A partial result is obtained by Kim [2], when part of the boundary of the domain is radially connected from a point , that is, is a connected interval. And there are some partial results on the higher-dimensional submanifold case (see [3, 4]). In case of , the problem remains open, even in the two-dimensional case (see [5]).

In this paper, we obtain two different type relative isoperimetric inequalities. First, using the modified volume introduced by Choe and Gulliver [6], we have a modified relative isoperimetric inequality in or without the curvature correct term:

(1.4)

where is the volume of a unit ball of , and is a domain of an -dimensional submanifold. In Theorem 2.11, (1.4) holds for lies on a geodesic sphere of or . In Theorem 2.3, (1.4) holds for , and is radially connected for a point .

Second, in Section 3 we obtain an inequality on usual volume for any minimal surface of a Riemannian manifold with sectional curvature bounded above by a constant :

(1.5)

But we cannot find a minimal surface which satisfies the equality. That is why we call (1.5) a weak relative isoperimetric inequality.

2. Modified Relative Isoperimetric Inequalities in a Space Form

We review the modified volume in and with constant sectional curvature and , respectively. Let be a point in the -dimensional sphere and let be the distance from to in .

Definition 2.1 (modified volume in ).

Given that is an -dimensional submanifold in , the modified volume of with center at is defined by

(2.1)

Embed in with being the north pole . For a domain, the geometric meaning of the modified volume of is the Euclidean volume of the orthogonal projection of into the counting orientation. Clearly, we have in

(2.2)

where is the usual volume of .

Similarly, let be a point in the -dimensional hyperbolic space and let be the distance from to in .

Definition 2.2 (modified volume in ).

Given that is an -dimensional submanifold in , the modified volume of with center at is defined by

(2.3)

Embed isometrically onto the hyperboloid in with the Minkowski metric such that is the point . Then for a domain, the modified volume equals the Euclidean volume of the projection of onto the hyperplane . Clearly, we have in

(2.4)

More precisely, see Choe and Gulliver's paper (see [6]).

Theorem 2.3.

Let be a closed convex set in . Assume that is a compact minimal surface in the outside such that is orthogonal to along . And is the distance from to and on . If is radially connected from the point , that is, is a connected interval, then one has

(2.5)

Equality holds if and only if is a totally geodesic half-disk with being a geodesic half-circle.

If is connected, then it is trivially radially connected from . If has two components, then using the same argument as [2, Corollary ] we obtain the following.

Corollary 2.4.

Let be a compact minimal surface satisfying the same assumptions as in Theorem 2.3 except the radially connectedness. Then the modified relative isoperimetric inequality in Theorem 2.3 holds if is connected or has two components that are connected by a component of

Before giving lemmas for proving Theorem 2.3, we define a cone. Given an -dimensional submanifold of or and a point in or , the -dimensional cone with center at is defined by the set of all minimizing geodesics from to a point of .

Lemma 2.5 (see [6, Lemma 4]).

• (a) If is an -dimensional minimal submanifold or a cone, and is the distance in from a fixed point, then

  

  (2.6)

where is the Laplacian on the submanifold and, in case is a cone, is the distance from the center of .

• (b) Suppose that is an -dimensional minimal submanifold or a cone. Then

(2.7)

Here again, in case of a cone , is the distance from the center of .

Proposition 2.6.

Let be a closed convex set in or . Assume that is an m-dimensional minimal submanifold in the outside such that is orthogonal to along and is the distance from to or . Let . In case of , one assumes that for all . Then one has

(2.8)

Proof.

Let and be the unit conomals to on and , respectively. By Lemma 2.5, we have

(2.9)

The makes the smallest angle with , that is, the unit normal vector to that lies in the two-dimensional plane spanned by and the tangent line of such that Clearly where is a unit tangent to . Since is a convex set, , and , we see that points outward of for every , where is the gradient in the . From the orthogonality condition, is a unit normal toward inside along the . So we have

(2.10)

for every , and

(2.11)

The similar proof holds for

Lemma 2.7 (see [6, Lemma 6]).

Let be Green's function of (, resp.), whose derivative is for ( for , resp.). If is an -dimensional minimal submanifold of (, resp.), then is subharmonic on (, resp.).

Now we estimate the angle of viewed from a point Recall the definition of the angle viewed from a point. For an -dimensional rectifiable set in and a point such that for all , the -dimensional angle of viewed from is defined by

(2.12)

where is the geodesic sphere of radius centered at , and the volume is measured counting multiplicity. Clearly, the angle does not depend on . There is obviously an analogous definition for the angle of viewed from .

Note that

(2.13)

where is the -dimensional density of at .

Proposition 2.8.

Let be an m-dimensional minimal submanifold satisfying the same assumptions as in Proposition 2.6. Then for any ,

(2.14)

Equality holds if and only if is totally geodesic and star shaped with respect to .

Proof.

Let be the geodesic ball centered at , radius , , and . By Lemma 2.7, we have subharmonic , where is distance from . Hence

(2.15)

where is outward unit conormal along the boundary.

Since, near , can be identified with totally geodesic half-sphere and and on as , we have

(2.16)

where is the unit conormal of the cone and the same argument holds as in Proposition 2.6.

Equality holds if and only if , , and , that is, is a star-shaped minimal cone with density at the center equal to . Since is the only -dimensional minimal submanifold in with volume , this completes the proof for . A similar proof holds for

Proof of Theorem 2.3.

We approach to the proof by comparison between and the cone . Since the cone is locally developable on a totally geodesic sphere , we reduce the proof to the proof of Theorem in [7] by doubling argument.

For each geodesic sphere centered at and radius , one has a local isometry between the curve and a great circle on . Hence we can develop on a great sphere ; one can find a curve (not necessarily closed) in and a local isometry from into , where is the north pole of . Clearly we have , , and Moreover, if we let and be the endpoints of , then .

We write , where , , is a connected component of . Note that may be empty or not.

If , then, after cutting along an appropriate geodesic and developing it onto the great sphere , may be identified with the cone in , where is a curve in given in terms of the polar coordinates by satisfying and Here, is the angle parameter of the cone. Now we define the doubling of by the doubling parametrization as follows:

(2.17)

If , then, after developing it onto , is identified with in , where is a curve in given in terms of the polar coordinates by defined on , where is the angle parameter of the cone. Choose such that Then we define the doubling of as follows:

(2.18)

In both cases, we have the following equalities:

(2.19)

Now let us define . Because of doubling process, we have

(2.20)

By Proposition 2.8, and then has a self-intersection point. By the geometric meaning of the modified volume, equals the Euclidean area of the standard projection of onto the plane containing the equator of . Let be the image of under the projection and the origin of the plane. Then , , but The last inequality arises from the fact that the projection is a length-shrinking map. Moreover, let and be the endpoints of , then . So we can apply [7, Lemma ] and conclude the following sharp isoperimetric inequality:

(2.21)

and equality holds if and only if is the boundary of a circle. So we have

(2.22)

By Proposition 2.6, we finally get

(2.23)

and equality holds if and only if and , or equivalently, is a totally geodesic half-disk centered at .

Proposition 2.9.

Let be a closed convex set in (, resp.) and let be an -dimensional compact minimal submanifold of (, resp.) satisfying that is orthogonal to along . Define for . Then (, resp.) is a monotonically nondecreasing function of for (, resp.), where is the -dimensional geodesic ball of radius centered in (, resp.).

Proof.

Let . is a convex set, , and the same argument holds as in Proposition 2.6; we have

(2.24)

Denote the volume forms on and by and , respectively. Then we have

(2.25)

It follows that

(2.26)

Hence we have

(2.27)

In the above inequality we used the fact that and on Therefore we obtain

(2.28)

This completes the proof for

The similar argument applies to

Remark 2.10.

The classical monotonicity of a minimal submanifold in the Euclidean or hyperbolic space can be found in [6, 8, 9].

In case lies on a geodesic sphere, automatically satisfies the radially connectivity. In this case, the relative isoperimetric inequality can be extended to the hyperbolic space, and to a minimal submanifold case (not necessarily a minimal surface case). More precisely, we have the following theorem.

Theorem 2.11.

Let be a closed convex set in or and let be an -dimensional compact minimal submanifold of or satisfying that is orthogonal to along . Assume that lies on a geodesic sphere centered at a point and that is the distance in or from . Furthermore, in case of , assume that . Then

(2.29)

where equality holds if and only if is an -dimensional totally geodesic half-ball centered at .

Proof.

Assume that is a minimal submanifold in with lying on a geodesic sphere. Let be the radius of the geodesic sphere. Since is a convex set and the same argument holds as in Proposition 2.6, then

(2.30)

By Proposition 2.9, Hence

(2.31)

and so the desired inequality follows. Equality holds if and only if is a cone with density at the center equal to , or equivalently, is a totally geodesic half-ball in . A similar proof holds for

3. Weak Relative Isoperimetric Inequalities in a Riemannian Manifold

The results in Section 2 are sharp but those require some extra assumptions on their boundary. And the results are concerned with the modified volume. In this section, by contrast, we obtain weak relative isoperimetric inequality which holds for any minimal surface and holds for the usual volume.

From now on, we denote as and as for simplicity.

Theorem 3.1.

Let be a closed convex set in a complete simply connected Riemannian manifold of sectional curvature bounded above by a constant . Assume that is a compact minimal surface in the outside of such that is orthogonal to along . In case of , one assumes that . Then one has

(3.1)

To prove the above theorem, we begin with the following lemmas on the Laplacian on functions of distance.

Lemma 3.2.

Let be an m-dimensional compact minimal submanifold in a simply connected Riemannian manifold of sectional curvature bounded above by a constant . Define for a fixed point .

(a)If , then one has on .

(b)If , then one has on .

(c)If , then one has on .

Proof.

See [6, 10] for the proof.

Lemma 3.3.

Let be a compact minimal surface satisfying the same assumptions as in Theorem 3.1. Define for any .

(a)If , then one has on

(1)

(b)If , then one has on

(2),

(3).

(c)If , then one has on

(4) if ,

(5) if .

Proof.

For , using Lemma 3.2(c), we have

(3.2)

Near , can be identified with totally geodesic half-sphere with constant sectional curvature . Since is a fundamental solution of Laplacian on , . Hence we obtain (4).

Next for (5) we compute

(3.3)

Note that

(3.4)

which proves (5) by (4).

For , using Lemma 3.2(a) and being a fundamental solution of Laplacian on , we get (1).

For , using Lemma 3.2(b) and being a fundamental solution of Laplacian on , we get (2). By similar argument as before, we get (3).

Proof of Theorem 3.1.

• (i) . Let be the distance from to . Integrating Lemma 3.3(3) over for the point , we get

  

  (3.5)

Since the same argument holds as in Proposition 2.6, we have

(3.6)

This inequality holds for all , and we can integrate it over and apply Fubini's theorem to obtain

(3.7)

By Lemma 3.2(b) and convexity of , we get

(3.8)

• (ii) . Integrate Lemma 3.3(1) twice and apply Lemma 3.2(a) as in (i).

• (iii) . Integrate Lemma 3.3(5) twice and apply Lemma 3.2(c) as in (i).

This completes the proof.