Mathematical Sciences: Applications of Analysis to Problems in Geometry

一名调查员将研究以下领域的问题： 黎曼流形的刚性定理， 流形没有共轭点，和尖锐的等周 不等式 第二名调查员将研究相关问题 在退化的偏微分方程组中， 方程，黎曼结构的等谱变形，以及 球面空间形式在球面中的极小等距嵌入。 本项目所要研究的问题来自于 一个例子，由于米尔诺尔的两个等谱流形，这是 不是等距的。 虽然最初的例子是16 三维的，最近三维的例子已经被 构建了 类似地，有人可能会问，如果两个圆形鼓， 有着相同的粗糙形状，敲击时发出的声音也是一样的， 都有同样的形状 事实上，他们必须。 这两 研究人员将研究相关的深层次问题， 维超曲面

One investigator will study problems in the areas of rigidity theorems for Riemannian manifolds, the structure of manifolds without conjugate points, and sharp isoperimetric inequalities. A second investigator will study related problems in the areas of degenerate systems of partial differential equations, isospectral deformations of Riemannian structures, and minimal isometric embeddings of spherical space forms in spheres. The problems to be investigated in this project arise from an example due to Milnor of two isospectral manifolds which are not isometric. While the original example was sixteen dimensional, more recently three dimensional examples have been constructed. Analogously, one might ask if two round drums which have the same rough shape and which sound the same when struck, have the same exact shape. In fact, they must. These two investigators will study related deep questions for higher dimensional hypersurfaces.