Mathematical Sciences: The Geometry and Analysis of CompleteManifolds

首席研究员将研究几何与完全流形分析之间的相互作用。本文将特别考虑非紧完全流形之间调和映射的存在性。研究者的早期工作以及Hamilton、Schoen、Yau、Liao和Tam的工作将为从单连通强负弯曲流形到Hadamard流形的调和映射的新研究奠定基础。研究者还将研究Yau的一个猜想，即具有有限拓扑类型的完全ricci -平坦Kaehler流形必须紧致于紧致Kaehler流形。广义曲面和它们之间的函数将由研究者详细研究。具体地说，他将集中研究某些利玛窦曲率范围内的曲面。要研究的曲面类型的一个例子是鞍形的无限延伸；这个表面具有这样的性质，在一个方向上，它向上弯曲，而在另一个方向上，它向下弯曲。这两个独立的曲率越大，里奇曲率就越大。

The principal investigator will study the interplay between geometry and analysis on complete manifolds. The existence of harmonic maps between non-compact complete manifolds will be given special consideration. Earlier work of the investigator as well as the work of Hamilton, Schoen, Yau, Liao, and Tam will serve as the basis for a new study of harmonic maps from simply connected strongly negatively curved manifolds into Hadamard manifolds. The investigator will also study one of Yau's conjectures that complete Ricci-flat Kaehler manifolds with finite topological type must be compactifiable to a compact Kaehler manifold. Generalized surfaces, and functions between them, will be studied in detail by the investigator. Specifically, he will concentrate on such surfaces within certain Ricci curvature ranges. An example of the kinds of surfaces to be studied would be the infinite extension of a saddle; this surface has the property that in one direction the surface curves away upward while in a different direction it curves away downward. The greater these two separate curvatures, the greater this Ricci curvature.