On the Positive Energy Theorem for Stationary Solutions to Fourth-Order Gravity

TL;DR

This paper proves a positive energy theorem for fourth-order gravitational theories, extending classical results in general relativity and connecting to geometric analysis problems like Q-curvature and rigidity phenomena.

Contribution

It introduces a positive energy theorem for higher-order gravity theories, linking gravitational energy positivity to geometric analysis and Q-curvature problems.

Findings

Established a positive energy theorem for fourth-order gravity

Connected positive mass theorems to Q-curvature and conformal geometry

Linked gravitational energy positivity to rigidity phenomena in geometric analysis

Abstract

In this paper we prove a positive energy theorem related to fourth-order gravitational theories, which is a higher-order analogue of the classical ADM positive energy theorem of general relativity. We will also show that, in parallel to the corresponding situation in general relativity, this result intersects several important problems in geometric analysis. For instance, it underlies positive mass theorems associated to the Paneitz operator, playing a similar role in the positive $Q$-curvature conformal prescription problem as the Schoen-Yau positive energy theorem does for the Yamabe problem. Several other links to $Q$-curvature analysis and rigidity phenomena are established.