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Prescribed Ricci curvature near an Einstein manifold with boundary

Erwann Delay

Abstract

Let (M, g) be a compact Einstein Riemannian manifold with boundary. We show that under certain conditions, the map that associates to a metric on M its Ricci curvature, its induced conformal class on the boundary, and its mean curvature on the boundary is locally invertible near g. The contravariant Ricci operator, as well as other operators such as the Einstein operator, are also studied.

Prescribed Ricci curvature near an Einstein manifold with boundary

Abstract

Let (M, g) be a compact Einstein Riemannian manifold with boundary. We show that under certain conditions, the map that associates to a metric on M its Ricci curvature, its induced conformal class on the boundary, and its mean curvature on the boundary is locally invertible near g. The contravariant Ricci operator, as well as other operators such as the Einstein operator, are also studied.

Paper Structure

This paper contains 7 sections, 11 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Definitions, Notations, and Conventions
  3. Fredholm Properties and Isomorphisms
  4. Case of Ricci Curvature
  5. Contravariant Ricci Operator
  6. Einstein-type Curvature
  7. Riemann-Christoffel Type Curvature

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth compact Einstein Riemannian manifold with boundary, with $\operatorname{Ric}(g)=\lambda g$. Suppose that $\lambda+\Lambda\neq0$ and that $-2\Lambda$ is not in the spectrum of the Hodge Laplacian acting on 1-forms with Dirichlet condition, nor in the spectrum of the Lichnerowi Moreover, the map $(r,[\gamma],\mathcal{H})\mapsto h$ is smooth from a neighborhood of $(0,[g^T], H

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4: ADN condition
  • Corollary 3.5
  • Remark 3.6
  • Proposition 4.1
  • proof
  • ...and 11 more