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 $$ \int_V R(g) dvol_g \ge 0 \text{\ \ and\ \ } I(V)\ge I(\mathbb{R}^n), $$ or $$ \int_V R(g) dvol_g \ge 0 \text{\ \ and\ \ } S(V)\ge S(\mathbb{R}^n),$$ 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. $\left(\cosh{\frac{d_g(p,x)}{2}}\right)^{-4}g$).