Research Interests: Differential Geometry, Geometric Analysis, Conformal Geometry and Mathematical Relativity
Vincent Bonini's research is in the fields of differential geometry and geometric analysis. He is particularly interested in the influence of curvature on global shape and the relationship between partial differential equations on manifolds and geometric problems. His recent research focuses on the application of analytic methods to asymptotically flat and asymptotically hyperbolic manifolds, which are significant in the mathematical theories of classic and quantum relativity. In the context of conformal geometry, his research in the asymptotically hyperbolic setting aims to strengthen the relation between the geometry and topology of certain conformally compact spaces and the geometry of their conformal infinities.
The primary aim of Vincent's recent research has been to advance the theory of asymptotically hyperbolic manifolds in relativity by further characterizing the geometric structure of the ends of asymptotically hyperbolic manifolds and exploring quasi-local notions of mass in the asymptotically hyperbolic setting. In a recent work, Bonini, Miao, and Qing used eigenfunction compactifications and a quasi-local mass characterization of Euclidean balls to establish a Ricci curvature rigidity result for weakly asymptotically hyperbolic manifolds without assuming spin structure. This article was published in June of 2006 in the journal Communications in Analysis and Geometry. More recently, Bonini and Qing established a Positive Mass Theorem on asymptotically hyperbolic manifolds with corners along a smooth hypersurface. The main analytic achievement in this work is the derivation of an asymptotic expansion of a solution to a perturbed eigenfunction equation from an integral representation of a solution, which allows them to understand the change of mass aspect tensor upon conformal change of metric. This article was published in the journal Annales Henri Poincaré in April 2008.
For an up-to-date list of publications, please see Vincent's Publications page.