Deformation of Scalar Curvature and Volume

TL;DR

This paper investigates the deformation of scalar curvature and volume on manifolds, providing conditions for stationary points, a local deformation theorem, and a gluing technique for constant scalar curvature metrics, with applications to counterexamples of Min-Oo's conjecture.

Contribution

It introduces a localized condition for stationary points of scalar curvature and volume, along with a deformation and gluing method for constructing metrics with prescribed scalar curvature and volume.

Findings

Established a local deformation theorem for scalar curvature and volume.

Developed a localized gluing theorem for constant scalar curvature metrics.

Constructed counterexamples with large volume and varied topology based on previous conjecture counterexamples.

Abstract

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo's conjecture such as that of Brendle-Marques-Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.