Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

Rodrigo Avalos; Paul Laurain; Nicolas Marque

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arXiv · 2204.03607

Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

Rodrigo Avalos·Paul Laurain·Nicolas Marque

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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.

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BibTeX RIS

Rodrigo Avalos, Paul Laurain, Nicolas Marque. 2022-04-07. Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces. https://arxiv.org/abs/2204.03607

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