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.
Jiayu Li
,
Shujing Pan
. A SURVEY ON THE ISOPERIMETRIC PROBLEM IN RIEMANNIAN MANIFOLDS[J]. Acta mathematica scientia, Series B, 2025
, 45(1)
: 228
-236
.
DOI: 10.1007/s10473-025-0118-6
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