Mathematical Sciences: The Geometry and Analysis of CompleteManifolds

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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.