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Rigidity of Einstein manifolds with positive Yamabe invariant

Letizia Branca, Giovanni Catino, Davide Dameno, Paolo Mastrolia

Abstract

We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant \emph{via} the $L^{\frac{n}{2}}$-norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions $5$ and $6$.

Rigidity of Einstein manifolds with positive Yamabe invariant

Abstract

We provide optimal pinching results on closed Einstein manifolds with positive Yamabe invariant in any dimension, extending the optimal bound for the scalar curvature due to Gursky and LeBrun in dimension four. We also improve the known bounds of the Yamabe invariant \emph{via} the -norm of the Weyl tensor for low-dimensional Einstein manifolds. Finally, we discuss some advances on an algebraic inequality involving the Weyl tensor for dimensions and .
Paper Structure (6 sections, 2 theorems, 101 equations, 1 figure, 9 tables)

This paper contains 6 sections, 2 theorems, 101 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Proof of Theorem \ref{['gianny']}
  4. Remarks on the sharp estimate for $Q$
  5. A lower bound in the six-dimensional case
  6. A numerical approach for the sharp constant in \ref{['qestimate']}

Key Result

Theorem 1.1

Let $(M,g)$ be a closed (conformally) Einstein manifold of dimension $n\geq 4$ with positive Yamabe invariant. Then, either $(M,g)$ is locally conformally flat (hence, a quotient of the round sphere) or, if $n\neq 5$ and $\operatorname{W}\not\equiv 0$, Moreover, equality holds in sharpinequality if and only if $(M,g)$ is locally symmetric. If $n=5$ and $\operatorname{W}\not\equiv 0$, then and eq

Figures (1)

  • Figure 1: Estimates for the order of convergence of ${\left|\dfrac{f(\operatorname{W})}{{\left|\operatorname{W}\right|}^3}-\dfrac{\sqrt{6}}{4}\right|}$ for $n=5$. Here, $\log(e_k)= \log{\left|\dfrac{f(\operatorname{W}_k)}{{\left|\operatorname{W}_k\right|}^3}-\dfrac{\sqrt{6}}{4}\right|}$, where $\operatorname{W}_k$ is the iteration at the $k$-th step. The scale of both axes are logarithmic.

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • proof : Proof of Corollary \ref{['corollary']}