Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

TL;DR

This paper establishes rigidity theorems for asymptotically Euclidean manifolds with specific curvature conditions, showing they must be Euclidean space under certain positivity assumptions, and introduces a fourth order Ricci-like tensor to analyze decay rates.

Contribution

It introduces a fourth order analogue to the Ricci tensor, $J_g$, and proves that Yamabe positive $J$-flat AE manifolds are isometric to Euclidean space, extending geometric analysis tools.

Findings

Yamabe positive $J$-flat AE manifolds are Euclidean

The $J_g$ tensor controls decay rates at infinity

A positive energy theorem for fourth order energy

Abstract

In this paper we prove some rigidity theorems associated to $Q$-curvature analysis on asymptotically Euclidean (AE) manifolds, which are inspired by the analysis of conservation principles within fourth order gravitational theories. A central object in this analysis is a notion of fourth order energy, previously analysed by the authors, which is subject to a positive energy theorem. We show that this energy can be more geometrically rewritten in terms of a fourth order analogue to the Ricci tensor, which we denote by $J_g$. This allows us to prove that Yamabe positive $J$-flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this $J$-tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of $J$ as a fourth order analogue to the Ricci tensor.