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Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

Rodrigo Avalos, Paul Laurain, Nicolas Marque

TL;DR

This work studies rigidity for fourth‑order curvature phenomena on asymptotically Euclidean (AE) manifolds, centering on $Q$-curvature and the tensor $J_g$ as a fourth‑order analogue of the Ricci tensor. The authors relate the fourth‑order energy $\,\mathcal{E}(g)$ to the asymptotic flux of the J‑Einstein tensor $G_J$ at infinity, establishing a conservation identity analogous to the ADM mass framework and leveraging a positive energy theorem. They prove that if $J_g=0$ and the Yamabe invariant $Y([g])>0$, then the AE manifold must be Euclidean, with a bootstrap mechanism showing that $J$‑flatness controls optimal decay via harmonic coordinates. Overall, the paper extends a Ricci‑flat rigidity paradigm to a fourth‑order, $Q$‑curvature–driven setting and provides precise decay control for geometric tensors on AE manifolds.

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.

Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

TL;DR

This work studies rigidity for fourth‑order curvature phenomena on asymptotically Euclidean (AE) manifolds, centering on -curvature and the tensor as a fourth‑order analogue of the Ricci tensor. The authors relate the fourth‑order energy to the asymptotic flux of the J‑Einstein tensor at infinity, establishing a conservation identity analogous to the ADM mass framework and leveraging a positive energy theorem. They prove that if and the Yamabe invariant , then the AE manifold must be Euclidean, with a bootstrap mechanism showing that ‑flatness controls optimal decay via harmonic coordinates. Overall, the paper extends a Ricci‑flat rigidity paradigm to a fourth‑order, ‑curvature–driven setting and provides precise decay control for geometric tensors on AE manifolds.

Abstract

In this paper we prove some rigidity theorems associated to -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 . This allows us to prove that Yamabe positive -flat AE manifolds must be isometric to Euclidean space. As a by product, we prove that this -tensor provides a geometric control for the optimal decay rates at infinity. This last result reinforces the analogy of as a fourth order analogue to the Ricci tensor.
Paper Structure (5 sections, 23 theorems, 110 equations)

This paper contains 5 sections, 23 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Weighted spaces on AE manifolds
  3. Harmonic end coordinates
  4. $\mathcal{E}(g)$ as a function of the $J$-Einstein tensor, proof of Theorem \ref{['HerzlichThmIntro']}
  5. $J_g$ flat means flat

Key Result

Theorem A

Let $(M^n,g)$, be a smooth $W^{3,p}_{-\tau}$ AE manifold with $p>n \ge 3$ and $\tau>0$. If $J_g=0$ and $Y([g])>0$, then $(M^n,g)\cong (\mathbb{R}^n,\cdot)$.

Theorems & Definitions (44)

  • Theorem A
  • Corollary
  • Theorem 1.1: Positive Energy avalos2021positive
  • Theorem B
  • Theorem C
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • ...and 34 more