Limites de massas quasi-locais de esferas em variedades Riemannianas

Abstract

In this work, we approach some concepts related to Mathematical Relativity, in order to discuss results related to quasi-local mass limits of spheres in three-dimensional Riemannian manifolds, in the light of the article entitled \Large-sphere and smallsphere limits of the Brown-York mass" by authors X.-Q. Fan, Y. Shi and L.-F. Tam. Initially, we studied and used decay properties of some geometric objects in an asymptotically

at three-dimensional manifold in order to study the limit of the quasi-local Brown-York mass in coordinate spheres with su ciently large radius, concluding that this limit converges, in the in nite, for the Arnowitt-Deser-Misner (ADM) mass of the asymptotically at manifold in question. Subsequently, we studied expansions in normal coordinates of geometric structures in order to study the Brown-York mass for geodesic spheres with su ciently small radius, using these results to present quasi-local mass expansions, as well as the behavior of certain spherical volumes. We also study results of the same character for the quasi-local Hawking and isoperimetric masses, in addition to approaching applications of these expansions in the face of the positive mass theorem.