- Research Article
- Open Access
Relative Isoperimetric Inequality for Minimal Submanifolds in a Riemannian Manifold
© Juncheol Pyo. 2010
- Received: 16 April 2009
- Accepted: 21 January 2010
- Published: 26 January 2010
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.
- Riemannian Manifold
- Minimal Surface
- Isoperimetric Inequality
- Geodesic Ball
- Angle Parameter
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:
where equality holds if and only if is a geodesic disk (see ).
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
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
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 , 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 ).
In this paper, we obtain two different type relative isoperimetric inequalities. First, using the modified volume introduced by Choe and Gulliver , we have a modified relative isoperimetric inequality in or without the curvature correct term:
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 :
But we cannot find a minimal surface which satisfies the equality. That is why we call (1.5) a weak relative isoperimetric inequality.
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 ).
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
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 ).
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
More precisely, see Choe and Gulliver's paper (see ).
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.
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 .
Suppose that is an -dimensional minimal submanifold or a cone. Then
Here again, in case of a cone , is the distance from the center of .
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
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 .
where is the -dimensional density of at .
Equality holds if and only if is totally geodesic and star shaped with respect to .
where is outward unit conormal along the boundary.
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  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.
and equality holds if and only if and , or equivalently, is a totally geodesic half-disk centered at .
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.).
This completes the proof for
The similar argument applies to
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.
where equality holds if and only if is an -dimensional totally geodesic half-ball centered at .
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
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.
To prove the above theorem, we begin with the following lemmas on the Laplacian on functions of distance.
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 .
Let be a compact minimal surface satisfying the same assumptions as in Theorem 3.1. Define for any .
(a)If , then one has on
(b)If , then one has on
(c)If , then one has on
(4) if ,
(5) if .
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).
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).
. Integrate Lemma 3.3(1) twice and apply Lemma 3.2(a) as in (i).
. Integrate Lemma 3.3(5) twice and apply Lemma 3.2(c) as in (i).
This completes the proof.
This work was supported in part by KRF-2007-313-C00057.
- Choe J, Gulliver R: The sharp isoperimetric inequality for minimal surfaces with radially connected boundary in hyperbolic space. Inventiones Mathematicae 1992, 109(3):495–503.MATHMathSciNetView ArticleGoogle Scholar
- Kim I: Relative isoperimetric inequality and linear isoperimetric inequality for minimal submanifolds. Manuscripta Mathematica 1998, 97(3):343–352. 10.1007/s002290050107MATHMathSciNetView ArticleGoogle Scholar
- Choe J, Ghomi M, Ritore M: The relative isoperimetric inequality outside convex domains in . Calculus of Variations and Partial Differential Equations 2007, 29(4):421–429. 10.1007/s00526-006-0027-zMATHMathSciNetView ArticleGoogle Scholar
- Choe J, Ritore M: The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds. Journal für die Reine und Angewandte Mathematik 2007, 605: 179–191.MATHMathSciNetGoogle Scholar
- Bandle C: Isoperimetric Inequalities and Applications. Pitman, Boston, Mass, USA; 1980:x+228.MATHGoogle Scholar
- Choe J, Gulliver R: Isoperimetric inequalities on minimal submanifolds of space forms. Manuscripta Mathematica 1992, 77(2–3):169–189.MATHMathSciNetView ArticleGoogle Scholar
- Choe J: The isoperimetric inequality for a minimal surface with radially connected boundary. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze 1990, 17(4):583–593.MATHMathSciNetGoogle Scholar
- Anderson MT: Complete minimal varieties in hyperbolic space. Inventiones Mathematicae 1982, 69(3):477–494. 10.1007/BF01389365MATHMathSciNetView ArticleGoogle Scholar
- Simon L: Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis. Volume 3. Australian National University, Canberra, Australia; 1983.Google Scholar
- Choe J: The isoperimetric inequality for minimal surfaces in a Riemannian manifold. Journal für die Reine und Angewandte Mathematik 1999, 506: 205–214.MATHMathSciNetView ArticleGoogle Scholar
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