Minimal Surfaces in Geometric Variational Problems

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Differential geometry is a modern version of Euclidean geometry that studies shapes inside curved surfaces in any number of dimensions. Two key notions in differential geometry, besides "length" and "angle," are: "minimal surfaces," which generalize the concept of a straight line, and "curvature," which measures how a surface is bent. The early development of differential geometry is partly attributable to physics. In the early 20th century, Einstein formulated a geometric theory of gravitation, general relativity, asserting that we live in a curved four-dimensional world ("spacetime") where energy and mass are manifestations of the curvature of spacetime, and minimal surfaces are indicative of black hole boundaries. Nowadays, differential geometry and minimal surfaces are at the heart of several physical theories and active mathematical research directions. The principal investigator (PI) will study problems regarding minimal surfaces that are motivated by general relativity and by the van der Walls--Cahn--Hilliard theory of phase transitions for multicomponent alloy systems. This project will also support the proposer's efforts to promote student learning, inclusion, and training through summer schools, workshops, and conferences, as well as via expository articles and notes.This project spans three related active research areas of minimal surface theory in differential geometry. First, the PI will investigate the construction of minimal surfaces as limiting min-max phase transitions. This construction has been recently shown to exhibit certain desirable stability properties that led to the proof of the "multiplicity one min-max conjecture" in three dimensions by the PI and Chodosh. The PI will continue this program, to produce surfaces with different curvatures and in different dimensions, as well as to better understand geometric implications of stable phase transitions. Second, as a step toward understanding limits of manifolds with nonnegative scalar curvature, the PI will study smooth and non-smooth three- and four-dimensional manifolds with nonnegative scalar curvature via the inherent bending effects of their embedded minimal surfaces. Third, the PI will apply this study of bending effects of minimal surfaces to investigate conjectured relationships between different mass notions in general relativity, where time-symmetric initial data sets are precisely manifolds with nonnegative scalar curvature.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.