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Existence of solutions to the Lichnerowicz equation: a new proof

Romain Gicquaud

Abstract

We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.

Existence of solutions to the Lichnerowicz equation: a new proof

Abstract

We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.

Paper Structure

This paper contains 4 sections, 8 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Local Yamabe invariant and first conformal eigenvalue
  3. Existence of solutions to the Lichnerowicz equation
  4. Existence of solutions to the prescribed scalar curvature equation

Key Result

Proposition 2.1

The functional $G$ defined in eqDefG is sequentially weakly lower semi-continuous on $W^{1, 2}$: for every weakly converging sequence $(u_k)_k$, $u_k \rightharpoonup u_\infty,$ we have $\liminf_{k\to \infty} G(u_k) \geq G(u_\infty).$

Theorems & Definitions (15)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thmCompact']}
  • ...and 5 more