On local rigidity theorems with respect to the scalar curvature

TL;DR

This paper uses Ricci flow to establish local rigidity theorems for scalar curvature, isoperimetric constants, and logarithmic Sobolev inequalities, demonstrating conditions under which a metric must be Euclidean or spherical.

Contribution

It introduces new local rigidity results involving scalar curvature, isoperimetric constants, and logarithmic Sobolev inequalities using Ricci flow techniques, including rigidity related to Perelman's entropy.

Findings

Proved Euclidean rigidity under scalar curvature and isoperimetric conditions.

Established spherical and hyperbolic rigidity theorems with weighted metrics.

Connected local rigidity to Perelman's $ u$-entropy and curvature bounds.

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. $\left(\cosh{\frac{d_g(p,x)}{2}}\right)^{-4}g$).