论文信息 - Some Sharp Isoperimetric Theorems for Riemannian Manifolds

Some Sharp Isoperimetric Theorems for Riemannian Manifolds

We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using dif- ferential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.

Frank Morgan | F. Morgan | David L. Johnson

[1] T. Aubin. Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire , 1976 .

[2] Christopher B. Croke. A sharp four dimensional isoperimetric inequality , 1984 .

[3] F. Morgan. Clusters minimizing area plus length of singular curves , 1994 .

[4] H. Bray. The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature (thesis) , 1997, 0902.3241.

[5] M. Grayson. Shortening embedded curves , 1989 .

[6] William K. Allard,et al. On the first variation of a varifold , 1972 .

[7] Bruce Kleiner,et al. An isoperimetric comparison theorem , 1992 .

[8] Manuel Ritoré,et al. The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds , 1996 .

[9] J. Nash. The imbedding problem for Riemannian manifolds , 1956 .

[10] Frank Morgan,et al. The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature , 2000 .

[11] C. B. Allendoerfer,et al. The Gauss-Bonnet theorem for Riemannian polyhedra , 1943 .

[12] P. Pansu. Sur la régularité du profil isopérimétrique des surfaces riemanniennes compactes , 1998 .

[13] Renato H. L. Pedrosa,et al. Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and , 1999 .

[14] S. Seki. On the Curvatura Integra in a Riemannian Manifold , 1953 .