Mathematical Sciences: Investigating Global Differential Geometry

这两位高级研究员将研究曲率的Lp界与黎曼流形的几何和拓扑之间的关系。他们将致力于证明一个新的拓扑不变量，如果为零，则蕴含着Cheeger-Gromov意义下的F-结构。我们将研究流形的等周常数和曲率之间的关系。研究人员将推广Heintze-Karcher对测地线社区体积的估计。最短闭测地线的下界将是非常有用的。计算机将被用来计算收敛黎曼流形的一些非平凡的例子。标准球面处处具有正的高斯曲率。即使发生变形，平均曲率也保持为正值。对于两个孔的圆环来说，情况并非如此。无论其中一个如何变形，平均曲率都必须为负值。在更高的维度上，情况要复杂得多。其他类型的曲率，如利玛窦曲率，发挥了核心作用。微分几何中的一个中心问题，两位研究人员将解决的问题是，这些曲率的平均如何与诸如洞的数量等拓扑性质相关。

The two senior investigators will study the relationship between Lp bounds on curvature and the geometry and topology of a Riemannian manifold. They will work towards a proof that a new topological invariant, if zero, implies an F-structure in the sense of Cheeger-Gromov. The relationship between the isoperimetric constant and the curvature of a manifold will be investigated. The investigators will generalize the Heintze- Karcher estimate on the volume of a geodesic neighborhood. A lower bound on the shortest closed geodesic would be very useful. A computer will be used to compute some nontrivial examples of convergent Riemannian manifolds. The standard sphere has positive Gaussian curvature everywhere. Even when deformed, the average curvature remains positive. Such is not the case for the two-holed torus. Regardless of how one of these is deformed, the average curvature must be negative. In higher dimensions the picture is much more complicated. Other sorts of curvature, such as Ricci curvature, play the central roles. A central problem in differential geometry, which the two investigators will address, is how the averages of these curvatures relates to topological properties such as numbers of holes.