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An elliptic proof of the splitting theorems from Lorentzian geometry

Mathias Braun, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Clemens Sämann

Abstract

We provide a new proof of the splitting theorems from Lorentzian geometry, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity (non-uniformly) elliptic $p$-d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman Lorentzian splitting theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

An elliptic proof of the splitting theorems from Lorentzian geometry

Abstract

We provide a new proof of the splitting theorems from Lorentzian geometry, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity (non-uniformly) elliptic -d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman Lorentzian splitting theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

Paper Structure

This paper contains 11 sections, 16 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. History and results
  3. Overview of strategy
  4. Technical challenges
  5. Outlook
  6. Equi-semiconcavity of the Busemann limits
  7. $p$-harmonicity of the Busemann function
  8. A Bochner-Ohta identity of homogeneity $2p-2<0$
  9. Local splitting in Newman's setting
  10. Local splitting in Galloway's setting
  11. Global splitting

Key Result

Theorem 1

If a spacetime $(M,g)$ satisfies the strong energy condition contains an isometrically embedded (timelike) copy $\gamma$ of the Euclidean line ${\mathbf R}$, and is either (a) globally hyperbolic or (b) timelike geodesically complete, then $(M,g)$ is isometric to $({\mathbf R} \times S, dt^2 - h)$ where $(S,h)$ is a complete Riemannian manifold with nonnegati

Theorems & Definitions (36)

  • Theorem 1: Lorentzian splitting theorem
  • Theorem 2: Busemann functions are Lipschitz near the line
  • proof
  • Remark 3: Conic intersections
  • Lemma 4: Upper supports to approximate Busemann functions
  • proof
  • Proposition 5: Equi-semiconcavity of Busemann limits near the line
  • proof
  • Corollary 6: Unit gradients converge a.e. for Busemann limits
  • proof
  • ...and 26 more