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Optimal isoperimetric inequalities. (English) Zbl 0585.49030

We make precise and prove the following four heuristic statements:
1. Optimal isoperimetric inequality. Corresponding to each m-1 dimensional closed surface T in \(R^ n\) there is an m dimensional surface Q having T as boundary such that \(| Q| \leq \gamma (m)| T|^{m/(m-1)}\) with equality if and only if T is a standard round m-1 sphere (of some radius) and Q is the corresponding flat m disk. Here \(| Q|\) and \(| T|\) denote the areas in dimensions m and m-1 respectively, and the optimal isoperimetric constant \(\gamma\) (m) is defined by the required equality.
This inequality includes as a special case the classical fact that hypersurface sphere have the smallest area for the volume they enclose.
2. Optimal isoperimetric inequality for mappings. Suppose \({\mathcal N}\) is a compact m dimensional Riemannian manifold with boundary and f:\(\partial {\mathcal N}\to R^ n\) is any Lipschitz boundary mapping. Then f can be extended to the interior to give a Lipschitz mapping g:\({\mathcal N}\to R^ n\) such that \[ (mapping\quad area\quad of\quad g)\leq \gamma (m)(mapping\quad area\quad of\quad f)^{m/(m-1)}. \] 3. Area-mean curvature characterizations of standard spheres. Suppose V is an m-1 dimensional surface in \(R^ n\) without boundary. If the mean curvature vectors of V do not exceed in length those of a standard round m-1 sphere of unit radius, then the m-1 area of V (actually of the extreme points of V) is not less than the m-1 area of the standard unit m-1 sphere. Furthermore, equality holds if and only if V is such a standard round m-1 sphere.
4. Mean curvature regularity theorem. Suppose T is a (possibly highly singular) m dimensional measure theoretic surface in \(R^ n\) which is combinatorially a cycle. In case the densities of T can be increased uniformly by some positive number to obtain a surface with bounded mean curvatures (in the sense of distributions), then T is a Hölder continuously differentiable submanifold of \(R^{n+1}\) except possibly for a compact singular set of zero m dimensional measure.
5. Optimal isoperimetric inequality for real rectifiable currents. Suppose \(T=t(S,\theta,\xi)\) is an m-1 dimensional real rectifiable cycle in \(R^ n\). Then there is an m dimensional real rectifiable current Q in \(R^ n\) such that \(\partial Q=T\) and M(Q)\(\leq \gamma (m+1)M(T)S(T)^{1/(m-1)}\). Furthermore, equality holds if and only if T is a real number multiple of the current associated with a standard round m-1 sphere in \(R^ n\) of some radius.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
28A75 Length, area, volume, other geometric measure theory
58A25 Currents in global analysis
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