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.
where equality holds if and only if is a geodesic disk (see ).
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.
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 .
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
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
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 ).
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 .
Lemma 2.7 (see [6, Lemma 6]).
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 .
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 .
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.).
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.
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.
To prove the above theorem, we begin with the following lemmas on the Laplacian on functions of distance.
which proves (5) by (4).
This completes the proof.
This work was supported in part by KRF-2007-313-C00057.
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