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A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds

Bartosz Bieganowski, Pietro d'Avenia, Jacopo Schino, Daniel Strzelecki

TL;DR

The work analyzes a Lichnerowicz-type elliptic equation arising from the Einstein–scalar constraint equations on closed manifolds with nonconstant mean curvature. It reduces the coupled constraints to a single equation for the conformal factor $u$, which contains Sobolev-supercritical and singular terms when the mean curvature is non-CMC. The authors prove the existence of a positive weak (and under extra regularity, $C^{2,\alpha}$) solution using a sub-/supersolution framework and a monotone fixed-point argument, and they derive explicit nonexistence criteria (NE1–NE5) that preclude smooth positive solutions in certain coefficient regimes. These results illuminate solvability conditions for Einstein–scalar field constraints under non-CMC data and provide tools for controlling or obstructing solutions via the data.

Abstract

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.

A Lichnerowicz equation in the Einstein-scalar field theory on non-CMC closed manifolds

TL;DR

The work analyzes a Lichnerowicz-type elliptic equation arising from the Einstein–scalar constraint equations on closed manifolds with nonconstant mean curvature. It reduces the coupled constraints to a single equation for the conformal factor , which contains Sobolev-supercritical and singular terms when the mean curvature is non-CMC. The authors prove the existence of a positive weak (and under extra regularity, ) solution using a sub-/supersolution framework and a monotone fixed-point argument, and they derive explicit nonexistence criteria (NE1–NE5) that preclude smooth positive solutions in certain coefficient regimes. These results illuminate solvability conditions for Einstein–scalar field constraints under non-CMC data and provide tools for controlling or obstructing solutions via the data.

Abstract

In the paper, we prove the existence of a positive and essentially bounded solution to a Lichnerowicz equation in the Einstein-scalar field theory on a closed manifold with non-constant mean curvature. In particular, the non-constant mean curvature gives rise to supercritical terms in the equation, on top of singular ones. We employ a recent fixed-point argument, which involves sub- and supersolutions. Additionally, we provide several conditions on the coefficients in the equation that prevent the existence of positive classical solutions.
Paper Structure (3 sections, 8 theorems, 73 equations)

This paper contains 3 sections, 8 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Existence of a weak solution
  3. Nonexistence of smooth positive solutions

Key Result

Theorem 2.2

Suppose that (A:coeff_in_proper_spaces)--(A:essinf_of_h) are satisfied. Then eq:1 has a weak solution which is positive and bounded away from zero almost everywhere on $M$. If, moreover, the functions in (A:coeff_in_proper_spaces) are Hölder continuous, then this solution is of class ${\mathcal{C}}^

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 7 more