Research

My research lies at the intersection of differential geometry, partial differential equations, and mathematical general relativity. I am particularly interested in geometric PDEs arising in general relativity and conformal geometry, their regularity theory, and the geometry of asymptotically Euclidean manifolds.

My current research centers on the following interconnected themes.

Mathematical General Relativity

A substantial part of my research concerns the mathematical analysis of the Einstein equations and the geometry of isolated gravitational systems.

I am particularly interested in the Einstein constraint equations, the geometry of initial data sets, and elliptic systems arising from their conformal formulation. In particular, I am interested in the geometry and asymptotic structure of asymptotically Euclidean initial data sets, motivated by the study of isolated systems in general relativity.

I am especially interested in geometric and analytic conditions governing asymptotic invariants such as the ADM energy, momentum, center of mass, and angular momentum, as well as geometrically distinguished coordinate systems at spatial infinity. One focus of this research is the search for geometric conditions that provide optimal control of the asymptotic behavior. I have recently related this question to the regularity of certain conformal compactifications, a connection that motivates some of my ongoing research.

Higher-Order Geometric PDEs

Another direction of my research concerns higher-order geometric equations, particularly those associated with Q-curvature.

Together with J. H. Lira and N. Marque, I have shown that a family of fourth-order gravitational theories gives rise to a notion of energy canonically associated with Q-curvature. Together with P. Laurain and J. H. Lira, we established a positive energy theorem for this quantity, parallel to the classical positive mass theorem (PMT) of general relativity. This result, in turn, plays an analogous role in the study of the Q-curvature counterpart of the Yamabe problem.

This project has developed further into the analysis of rigidity phenomena for fourth-order geometric PDEs, including the study of Q-singular spaces, and their use in controlling both the asymptotic behavior and regularity of asymptotically Euclidean manifolds.

Geometric Analysis and Conformal Geometry

I am interested in the analysis of geometric partial differential equations arising in conformal geometry, with particular emphasis on the Yamabe problem and analogous problems associated with higher-order conformal invariants such as Q-curvature.

More recently, I have become interested in low-regularity versions of the Yamabe problem and, more broadly, in what might be called “geometric elliptic regularity” in conformal geometry. By this, I mean understanding which conformal objects determine the regularity of the best representative of a conformal class, as well as exploring applications of such regularity theorems to geometric analysis and mathematical general relativity.

Current Research

My current and ongoing projects include regularity problems for conformal structures on closed manifolds, conformal Green functions and Yamabe metrics of Sobolev regularity, the asymptotic geometry of isolated systems at spatial infinity, and regularity questions for the coupled Einstein constraint equations.