On local rigidity theorems with respect to the scalar curvature

Abstract

By using the Ricci flow, we study local rigidity theorems regarding scalar curvature, isoperimetric constant and best constant of $L^2$ logarithmic Sobolev inequality. Precisely, we prove that if a metric $g$ on an open set $V$ in an $n$-dimensional Riemannian manifold satisfies $${\int }_{V}R(g)dvo{l}_g\ge 0\text{ and }I(V)\ge I({\mathbb{R}}^n),$$ or $${\int }_{V}R(g)dvo{l}_g\ge 0\text{ and }S(V)\ge S({\mathbb{R}}^n),$$ then $g=g_{{\mathbb{R}}^n}$ on $V$, where $R(g)$ is the scalar curvature of $g$, ${\mathbb{R}}^n$ is Euclidean space, $I(V)$ is the isoperimetric constant of $V$ and $S(V)$ is best constant of $L^2$ logarithmic Sobolev inequality of $V$. Moreover,we also obtain the local ${\mathbb{R}}^n$-rigidity about local Perelman's $\nu$-entropy, and local ${\mathbb{S}}^n$-rigidity (resp. ${\mathbb{H}}^n$-rigidity) theorems regarding the cases concerning $R(g)\ge n(n-1)$ (resp. $R(g)\ge -n(n-1)$), weighted isoperimetric constant and best constant of weighted $L^2$ logarithmic Sobolev inequality for the weighted metric ${\left(\cos \frac{d_g(p,x)}{2}\right)}^{-4}g$ (resp. ${\left(\cosh \frac{d_g(p,x)}{2}\right)}^{-4}g$).

arXiv commentWe also study corresponding rigidity theorems in the new version for cases where the scalar curvature is bounded below by $-n(n-1)$ or $n(n-1)$. Precisely, we add Theorem 1.7, Theorem 1.8 and add Section 5 for the proofs