Near-Constant Mean Curvature Solutions of the Einstein Constraint Equations with Non-Negative Yamabe Metrics

TL;DR

This paper demonstrates that under certain conditions, conformal data with non-negative Yamabe metrics lead to solutions of the Einstein constraint equations, extending previous results to broader metric classes.

Contribution

It extends the existence results for Einstein constraint solutions to include positive and zero Yamabe class metrics with small mean curvature gradients.

Findings

Solutions exist for conformal data with non-negative Yamabe metrics.

The results generalize previous work by removing the nonzero mean curvature requirement.

Applicable to closed manifolds with positive or zero Yamabe class metrics.

Abstract

We show that sets of conformal data on closed manifolds with the metric in the positive or zero Yamabe class, and with the gradient of the mean curvature function sufficiently small, are mapped to solutions of the Einstein constraint equations. This result extends previous work which required the conformal metric to be in the negative Yamabe class, and required the mean curvature function to be nonzero.