Dirichlet problem for the constant mean curvature equation and CMC foliation in the extended Schwarzschild spacetime

TL;DR

This paper proves the existence and uniqueness of solutions to the constant mean curvature equation in Schwarzschild spacetime and resolves the CMC foliation conjecture from 2003.

Contribution

It establishes the Dirichlet problem for spacelike, spherically symmetric CMC equations and confirms the CMC foliation conjecture in the extended Schwarzschild spacetime.

Findings

Existence and uniqueness of solutions to the Dirichlet problem

Complete resolution of the CMC foliation conjecture

Application to spherically symmetric spacelike hypersurfaces

Abstract

We prove the existence and uniqueness of the Dirichlet problem for spacelike, spherically symmetric, constant mean curvature equation with symmetric boundary data in the extended Schwarzschild spacetime. As an application, we completely solve the CMC foliation conjecture which is posted by Malec and O Murchadha in 2003.