Mathematical Sciences: Applications in the Geometric Calculus of Variations

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9626560 Pitts The project lies in the area of minimal surface theory. The investigator plans to utilize the geometric calculus of variations to consider desingularization and construction of minimal surfaces. In addition, the investigator proposes to show that any compact Riemannian manifold contains infinitely many distinct minimal hypersurfaces, again using the variational approach. Minimal surfaces first arose in a soap film experiment done by the Belgian physicist Joseph Plateau around 1850, where Plateau observed that a soap film surface bounded by a closed wire contour has the least surface area amongst all surfaces spanned by the same wire contour. Minimal surfaces turn out to have other remarkable mathematical properties (the average curvature at any point must equal zero); their study involves several branches of modern mathematics. Additionally, minimal surfaces can be used to model condensed matter, e.g., compound polymer systems.