Rigidity, Stability, Regularity, and Resolution Theorems in the Geometric Calculus of Variations

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This project investigates a set of mathematical models used in areas of Physics like General Relativity or surface tension theory. The focus is on aspects of these models that require the development of new mathematical tools and ideas. This kind of basic research advances Mathematics while keeping it grounded in the natural sciences. An important component of the project is research training, both at the graduate and the postdoctoral level.Soap films are classically modeled as two-dimensional surfaces, an approach that is correct in first approximation, but prevents the understanding of some of their physical properties such as bursting or bulging. The investigator has recently initiated the study of soap-films in the context of capillarity theory (soap films as three-dimensional regions with positive volume), and this project will focus on several new problems challenging the boundaries of GMT (Geometric Measure Theory) and the Calculus of Variations and aimed at understanding the interplay between the volume constraint and geometric features like collapsing and thickness. The investigator also intends to reformulate the soap film capillarity model in the context of diffused interface capillarity, thus adding a second length scale needed to further inquiry into collapsing phenomena. Diffused interface capillarity is also used to propose a PDE (Partial Differential Equations) approach to long standing questions concerning the volume-preserving mean curvature flow. PDE and GMT methods are central to the part of the project where the investigator will undertake a systematic analysis of a new kind of geometric regularity theorems (mesoscale flatness criteria) and their applications to the study of large-volume exterior isoperimetry and stable constant mean curvature foliations at infinity in General Relativity. Finally, novel approaches to natural generalizations of the Yamabe and Kazdan–Werner problems with boundary in Riemannian Geometry will also be studied as part of the project using methods from the theory of Optimal Mass Transport.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.