Mathematical Sciences: Applications in the Geometric Calculus of Variations

9626560皮茨 该项目属于最小曲面理论领域。研究人员计划利用几何变分法来考虑最小曲面的去奇异化和构造。此外，研究者建议证明任何紧致黎曼流形包含无穷多个不同的极小超曲面，再次使用变分方法。 最小表面最早出现在1850年左右由比利时物理学家约瑟夫·普拉托（Joseph Plateau）所做的肥皂膜实验中，其中普拉托观察到，由封闭的金属丝轮廓所包围的肥皂膜表面在由相同金属丝轮廓所跨越的所有表面中具有最小的表面积。极小曲面还有其他显著的数学性质（任意点的平均曲率必须等于零）;它们的研究涉及到现代数学的几个分支。此外，最小表面可用于模拟凝聚态物质，例如，复合聚合物体系

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.