A SURVEY ON THE ISOPERIMETRIC PROBLEM IN RIEMANNIAN MANIFOLDS

doi:10.1007/s10473-025-0118-6

Abstract

Abstract: This is a survey of the results in [14] regarding the isoperimetric problem in the Riemannian manifold. We consider a mean curvature type flow in the Riemannian manifold endowed with a non-trivial conformal vector field, which was firstly introduced by Guan and Li [8] in space forms. This flow preserves the volume of the bounded domain enclosed by a star-shaped hypersurface and decreases the area of hypersurface under certain conditions. We will prove the long time existence and convergence of the flow. As a result, the isoperimetric inequality for such a domain is established.

Key words: conformal vector fields, isoperimetric inequality, mean curvature type flow

CLC Number:

53C18

Jiayu Li, Shujing Pan. A SURVEY ON THE ISOPERIMETRIC PROBLEM IN RIEMANNIAN MANIFOLDS[J].Acta mathematica scientia,Series B, 2025, 45(1): 228-236.

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References

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