Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new approach towards the construction of initial data in general relativity with positive Yamabe invariant and arbitrary mean curvature

Armand Coudray, Romain Gicquaud

Abstract

This paper revisits the classical construction of initial data using the conformal method, as originally proposed by Holst, Nagy, and Tsogtgerel and later refined by Maxwell. We demonstrate that the existence of the solution can be proven using the Banach fixed point theorem, whereas the original proof relied on the Schauder fixed point theorem. This new approach has two main advantages: it guarantees the uniqueness of the solution to the equations of the conformal method as soon as one imposes a bound on the physical volume of it and it provides an explicit construction of the solution.

A new approach towards the construction of initial data in general relativity with positive Yamabe invariant and arbitrary mean curvature

Abstract

This paper revisits the classical construction of initial data using the conformal method, as originally proposed by Holst, Nagy, and Tsogtgerel and later refined by Maxwell. We demonstrate that the existence of the solution can be proven using the Banach fixed point theorem, whereas the original proof relied on the Schauder fixed point theorem. This new approach has two main advantages: it guarantees the uniqueness of the solution to the equations of the conformal method as soon as one imposes a bound on the physical volume of it and it provides an explicit construction of the solution.
Paper Structure (5 sections, 12 theorems, 115 equations)

This paper contains 5 sections, 12 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Estimates for the Lichnerowicz equation
  3. Estimates for the vector equation
  4. Estimates for the solution under a volume bound
  5. Existence and uniqueness of the solution

Key Result

Theorem 1

Let $(M, g)$ be a compact Riemannian manifold, $\tau$ a given function on $M$ and $\sigma$ a non-zero TT-tensor for $g$ satisfying the regularity assumptions stated above. Let $V_{\max}$ be a small enough positive constant and $\omega_0 > 0$. Then there exists a constant $c = c(M, g, \tau, \omega_0, there exists a unique solution $(\varphi, W) \in W^{2, p}(M, \mathbb{R}) \times W^{2, p}(M, TM)$ to

Theorems & Definitions (22)

  • Theorem 1
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 12 more