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<p>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.</p>

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A survey on the isoperimetric problem in Riemannian manifolds

  • 期刊论文

  • 发表日期：

    2025年1月24日

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  • Jiayu Li,
  • Shujing Pan

摘要

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.