V-static spaces with positive isotropic curvature

TL;DR

This paper classifies critical metrics of the volume functional on compact manifolds with boundary and positive isotropic curvature, identifying when such manifolds are isometric to standard geometric models like spheres or products.

Contribution

It provides a complete classification of critical metrics under specific curvature conditions, extending understanding of geometric structures with positive isotropic curvature.

Findings

Manifolds with positive isotropic curvature satisfying the critical metric condition are isometric to geodesic balls in spheres when 

They are isometric to hemispheres or products of an interval with a sphere when 

The classification covers all cases under the given curvature and boundary conditions.

Abstract

In this paper, we give a complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $\partial M$ having positive isotropic curvature. We prove that for a pair $(f, \kappa)$ of a nontrivial smooth function $f: M \to {\Bbb R}$ and a nonnegative real number $\kappa$, if $(M, g)$ having positive isotropic curvature satisfies $$ Ddf - (\Delta f)g - f{\rm Ric} = \kappa g, $$ then $(M, g)$ is isometric to a geodesic ball in ${\Bbb S}^n$ when $\kappa >0$, and either $M$ isometric to ${\Bbb S}^n_+$, or the product $I \times {\Bbb S}^{n-1}$, up to finite cover when $\kappa =0$.