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The power series expansions of logarithmic Sobolev, $\mathcal{W}$- functionals and scalar curvature rigidity

Liang Cheng

TL;DR

The paper addresses rigidity phenomena under scalar curvature lower bounds and isoperimetric profile comparisons by deriving power series expansions of logarithmic Sobolev and Perelmans \(\boldsymbol{\mu}\)-functional with carefully constructed test functions. The main technique yields explicit second-order curvature terms, enabling a local-to-global rigidity: if $\operatorname{Sc}(x) \ge n(n-1)K$ on an open set $V$ and $\operatorname{I}(V,\beta) \ge \operatorname{I}(M^n_K,\beta)$ for small $\beta$, then the sectional curvature satisfies $\operatorname{Sec}(x)=K$ on $V$, complemented by new rigidity results for the logarithmic Sobolev inequality and Perelmans $\boldsymbol{\mu}$-functional. The approach links sharp functional-inequality data to curvature constraints via local expansions and Schwarz symmetrization, offering a novel route to isoperimetric and scalar-curvature rigidity. These results extend classical Bishop-Gromov-type rigidity to scalar curvature settings and provide tools for analyzing rigidity in geometric analysis contexts.

Abstract

In this paper, we obtain that the logarithmic Sobolev and $\mathcal{W}$-functionals admit remarkable power series expansions when appropriate test functions are selected. Using these expansions formulas, we prove that for an open subset $V$ in an $n$-dimensional manifold $M$ with $\bar{V}\subset M$ satisfying: (a)The scalar curvature of $V$ satisfies the lower bound:$$\operatorname{Sc}(x) \geq n(n-1)K \quad \text{for all } x \in V,$$ (b) The isoperimetric profile of $V$ is no less than that of space form $M^n_K$:$$ \operatorname{I}(V,β) := \inf_{\substack{Ω\subset V \\ \mathrm{Vol}(Ω)=β}} \mathrm{Area}(\partial Ω) \geq \operatorname{I}(M^n_K,β) \quad \text{for some } β_0>0 \text{ and all } 0<β<β_0,$$\textbf{then} the sectional curvature of $V$ must satisfy $$\operatorname{Sec}(x) = K \quad \text{for all } x \in V.$$ Additionally, we derive some new scalar curvature rigidity theorems concerninglogarithmic Sobolev inequality and Perelman's $\boldsymbolμ$-functional.

The power series expansions of logarithmic Sobolev, $\mathcal{W}$- functionals and scalar curvature rigidity

TL;DR

The paper addresses rigidity phenomena under scalar curvature lower bounds and isoperimetric profile comparisons by deriving power series expansions of logarithmic Sobolev and Perelmans -functional with carefully constructed test functions. The main technique yields explicit second-order curvature terms, enabling a local-to-global rigidity: if on an open set and for small , then the sectional curvature satisfies on , complemented by new rigidity results for the logarithmic Sobolev inequality and Perelmans -functional. The approach links sharp functional-inequality data to curvature constraints via local expansions and Schwarz symmetrization, offering a novel route to isoperimetric and scalar-curvature rigidity. These results extend classical Bishop-Gromov-type rigidity to scalar curvature settings and provide tools for analyzing rigidity in geometric analysis contexts.

Abstract

In this paper, we obtain that the logarithmic Sobolev and -functionals admit remarkable power series expansions when appropriate test functions are selected. Using these expansions formulas, we prove that for an open subset in an -dimensional manifold with satisfying: (a)The scalar curvature of satisfies the lower bound: (b) The isoperimetric profile of is no less than that of space form :\textbf{then} the sectional curvature of must satisfy Additionally, we derive some new scalar curvature rigidity theorems concerninglogarithmic Sobolev inequality and Perelman's -functional.
Paper Structure (4 sections, 9 theorems, 125 equations)

This paper contains 4 sections, 9 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Power series expansion formulas of logarithmic Sobolev and $\mathcal{W}$-functionals
  3. proofs of Theorem \ref{['mu_rigidity_extension']} and Theorem \ref{['mu_rigidity']}
  4. the proof of Theorem \ref{['rigidity_iso_profile']}

Key Result

Theorem 1.1

Let $(M^n,g)$ be an $n$-dimensional Riemannian manifold, and let $V$ be an open subset with $\overline{V}\subset M$. Suppose that the following two conditions hold: (a) The scalar curvature of $V$ satisfies (b) There exists $\beta_0 > 0$ such that the isoperimetric profile of $V$ satisfies where $M^n_K$ is the space form of constant sectional curvature $K$. Then the sectional curvature of $V$ sa

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • ...and 12 more