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Rigidity Results Involving Stabilized Scalar Curvature

Yipeng Wang

TL;DR

The paper studies rigidity phenomena for $T^{\rtimes}$-stabilized scalar curvature, building on Brendle–Hung's free-boundary systolic inequality. It develops foliations by free boundary weighted constant mean curvature hypersurfaces and introduces a monotone quantity from Ricci flow coupled with a heat equation to drive rigidity; it extends both free-boundary and closed-surface systolic inequalities to the stabilized curvature setting and establishes generalized Geroch-type rigidity results, including spin cases. The work provides inductive rigidity arguments yielding universal-cover product structures (often with flat factors) under equality, and establishes a monotone evolution law for the stabilized curvature, connecting to Perelman-type functionals. Together, these results deepen the link between systolic-type inequalities, stabilized curvature notions, and rigidity phenomena in low dimensions, with implications for geometric decomposition and flatness in the stabilized framework.

Abstract

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of \(T^{\rtimes}\)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the \(T^{\rtimes}\)-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.

Rigidity Results Involving Stabilized Scalar Curvature

TL;DR

The paper studies rigidity phenomena for -stabilized scalar curvature, building on Brendle–Hung's free-boundary systolic inequality. It develops foliations by free boundary weighted constant mean curvature hypersurfaces and introduces a monotone quantity from Ricci flow coupled with a heat equation to drive rigidity; it extends both free-boundary and closed-surface systolic inequalities to the stabilized curvature setting and establishes generalized Geroch-type rigidity results, including spin cases. The work provides inductive rigidity arguments yielding universal-cover product structures (often with flat factors) under equality, and establishes a monotone evolution law for the stabilized curvature, connecting to Perelman-type functionals. Together, these results deepen the link between systolic-type inequalities, stabilized curvature notions, and rigidity phenomena in low dimensions, with implications for geometric decomposition and flatness in the stabilized framework.

Abstract

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of -stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the -stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.
Paper Structure (8 sections, 37 theorems, 130 equations)

This paper contains 8 sections, 37 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. The Free Boundary Systolic Inequality
  3. The Equality Case of Theorem \ref{['thm:free-boundary']}
  4. The Systolic Inequality
  5. The Equality Case of Theorem \ref{['thm:Zhu']}
  6. Proof of Theorem \ref{['thm:generalized-Geroch']}
  7. Proof of Theorem \ref{['thm:generalized-Geroch-spin']}
  8. Evolution of Stabilized Scalar Curvature under Ricci Flow Coupled with the Heat Equation

Key Result

Theorem 1.1

Let $(M^n, g)$ be a compact, connected, orientable Riemannian manifold of dimension $n \leq 7$ with nonempty boundary $\partial M$. Denote by $\eta$ the outward-pointing unit normal vector along $\partial M$. Let $\varphi$ be a smooth function defined on $M$. Suppose there is a smooth map $\xi: \par Assume that at point point in $M$. Then we have Moreover, equality occurs if and only if $\varphi

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Gromov--Lawson GL-1, GL-2; Schoen--Yau SY-3manifold
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Theorem 2.2: Brendle--Hung Brendle-Hung
  • proof
  • Corollary 2.3
  • ...and 50 more