2022 | Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces | R. Avalos, P. Laurain, N. MarqueLink zum Preprint
Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces
Autoren: R. Avalos, P. Laurain, N. Marque
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.
2022 | Einstein Type Systems on Complete Manifolds | R. Avalos, J. Lira, N. MarqueLink zum Preprint
Einstein Type Systems on Complete Manifolds
Autoren: R. Avalos, J. Lira, N. Marque
In the present paper, we study the coupled Einstein Constraint Equations (ECE) on complete manifolds through the conformal method, focusing on non-compact manifolds with flexible asymptotics. In particular, we do not impose any specific model for infinity. First, we prove an existence criteria on compact manifolds with boundary which applies to more general systems and can be seen as a natural extension of known existence theory for the coupled ECE. Building on this, we prove an L^p existence based on existence of appropriate barrier functions for a family of physically well-motivated coupled systems on complete manifolds. We prove existence results for these systems by building barrier functions in the bounded geometry case. We conclude by translating our result to the H^s formulation, making contact with classic works. To this end, we prove several intermediary L^2 regularity results for the coupled systems in dimensions n \le 12 which fills certain gaps in current initial data analysis.
2021 | Energy in Fourth Order Gravity | Rodrigo Avalos, Jorge H. Lira, Nicolas MarqueLink zum Preprint
Energy in Fourth Order Gravity
Autoren: Rodrigo Avalos, Jorge H. Lira, Nicolas Marque
In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this notion of energy which allows us to interpret it, and introduce several study cases where its analysis seems tractable. Finally, positive energy theorems are presented in restricted situations.
2020 | Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications | Yann Bernard, Paul Laurain, Nicolas MarqueLink zum Preprint
Energy Estimates for the Tracefree Curvature of Willmore Surfaces and Applications
Autoren: Yann Bernard, Paul Laurain, Nicolas Marque
We prove an ϵ-regularity result for the tracefree curvature of a Willmore surface with bounded second fundamental form. For such a surface, we obtain a pointwise control of the tracefree second fundamental form from a small control of its L2-norm.Several applications are investigated. Notably, we derive a gap statement for surfaces of the aforementioned type. We further apply our results to deduce regularity results for conformal minimal spacelike immersions into the de Sitter space S4,1.