Besse conjecture with positive isotropic curvature

Abstract

The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function $f$ on the manifold that satisfies the following $$(1+f)\mathrm{Ric}=Ddf+\frac{nf+n-1}{n(n-1)}sg.$$ It has been conjectured that if $(g,f)$ is a solution of the critical point equation, then $g$ is Einstein and so $(M,g)$ is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.

21 pages without figures