Mathematical Sciences: Theoretical and Numerical Existence of Many Minimal Surfaces on Riemannian Manifolds

首席研究员将研究极小超曲面 在黎曼流形中。他将继续开发和应用 几何测度理论和几何 极小解存在性问题的变分法 表面。此外，主要研究者将继续 开发构造极小值的数值方法。 该奖项支持的研究将试图 描述曲面在弯曲空间中的位置。最小 表面是具有最小面积的表面， 在曲面上绘制的边界曲线。这些表面经常 在自然界中发生，因为许多过程都是根据一些 最小能量原理一个重要的例子是固体- 凝固过程中的液体边界。因为有这么 物理学中几何最小化的许多重要例子 科学，对理论方面的透彻理解， 预计这一进程将对科学产生广泛影响。

The principal investigator will study minimal hypersurfaces in Riemannian manifolds. He will continue to develop and apply techniques from geometric measure theory and the geometric calculus of variations to existence problems for minimal surfaces. In addition the principal investigator will continue to develop numerical methods for constructing minimizers. The research supported by this award will attempt to describe how minimal surfaces lie in curved space. Minimal surfaces are surfaces which have minimal area for any prescribed boundary curve drawn on the surface. These surfaces frequently occur in nature because many processes develop according to some minimum energy principle. An important example is the solid- liquid boundary in a solidification process. Because there are so many important examples of geometric minimization in the physical sciences, a thorough understanding of the theoretical aspects of the process can be expected to have a broad impact on science.