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New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals

Marco Bernardini

Abstract

We prove a new rigidity result for metrics defined on closed smooth $ n $-manifolds that are critical for the quadratic functional $ \mathfrak{F}_{t} $, which depends on the Ricci curvature $ Ric $ and the scalar curvature $ R $, and that satisfy a pinching condition of the form $ Sec > εR $, where $ ε$ is a function of $ t $ and $ n $, while $ Sec $ denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying $ Sec > \frac{1}{48} R $ are Einstein and, by a known result, are isometric to $ \mathbb{S}^{4} $, $ \mathbb{RP}^{4} $ or $ \mathbb{CP}^{2} $.

New Rigidity Results for Critical Metrics of Some Quadratic Curvature Functionals

Abstract

We prove a new rigidity result for metrics defined on closed smooth -manifolds that are critical for the quadratic functional , which depends on the Ricci curvature and the scalar curvature , and that satisfy a pinching condition of the form , where is a function of and , while denotes the sectional curvature. In particular, we show that Bach-flat metrics with constant scalar curvature satisfying are Einstein and, by a known result, are isometric to , or .
Paper Structure (3 sections, 10 theorems, 50 equations)

This paper contains 3 sections, 10 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. The Euler-Lagrange equations and some Lemmas
  3. Proof of Theorem \ref{['th:Io_1']}, Theorem \ref{['th:Io_2']}

Key Result

Theorem 1

Let $M^{n}$ closed smooth n-manifold with $n \geq 3$, let $g$ be a critical metric for $\mathfrak{F}_{t}$ on $\mathcal{M}_{1}(M^{n})$ with $t \leq - \frac{1}{2}$, such that $R \geq 0$ and $Sec > \frac{1 + 2 t}{(n-2)^{2}} R$. Then, $g$ is Einstein.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • proof
  • Proposition 1
  • Corollary 3
  • Corollary 4
  • Proposition 2
  • Proposition 3
  • ...and 1 more