Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scalar curvature rigidity of the four-dimensional sphere

Simone Cecchini, Jinmin Wang, Zhizhang Xie, Bo Zhu

Abstract

Let $(M,g)$ be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by $n(n-1)$. In this paper, we prove that if $f$ is a smooth map of non-zero degree from $(M, g)$ to the unit four-sphere, then $f$ is an isometry. Following ideas of Gromov, we use $μ$-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.

Scalar curvature rigidity of the four-dimensional sphere

Abstract

Let be a closed connected oriented (possibly non-spin) smooth four-dimensional manifold with scalar curvature bounded below by . In this paper, we prove that if is a smooth map of non-zero degree from to the unit four-sphere, then is an isometry. Following ideas of Gromov, we use -bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case of strict inequality. Our proof of rigidity is based on the harmonic map heat flow coupled with the Ricci flow.
Paper Structure (5 sections, 15 theorems, 50 equations)

This paper contains 5 sections, 15 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. $\mu$-bubbles with mean curvature bound
  3. scalar-mean bounds on spherical bands
  4. Ricci flow, harmonic map heat flow, and Einstein metrics
  5. Rigidity of the four-dimensional sphere

Key Result

Theorem 1.1

Let $(M,g)$ be an $n$-dimensional closed connected spin Riemannian manifold with $\mathop{\mathrm{Sc}}\nolimits_g\geq n(n-1)$. If $f\colon(M,g)\to(\mathbb{S}^n,g_{\mathbb{S}^n})$ is a smooth, distance non-increasing map of non-zero degree, then $f$ is an isometry.

Theorems & Definitions (27)

  • Theorem 1.1: Llarull
  • Theorem A
  • Theorem 1.2: Listing:2010te
  • Lemma 2.1: zhu_width_mu, Raede23
  • Lemma 2.2: zhu_width_mu,Raede23
  • Remark 2.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 17 more