Scalar Curvature and Projective Compactness

TL;DR

This paper shows that under mild conditions, if a metric's projective structure extends to the boundary but its Levi-Civita connection does not, then the metric is projectively compact of order 2, with implications for scalar curvature and boundary conformal structure.

Contribution

It introduces a new interpretation of scalar curvature via projective geometry and establishes conditions under which metrics are projectively compact of order 2.

Findings

Metrics with extending projective structure have scalar curvature extending smoothly.

Such metrics are projectively compact of order 2, relating volume growth to boundary behavior.

The work links scalar curvature, projective compactness, and boundary conformal structures.

Abstract

Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.