Download PDF of Lectures on Differential Geometry by Richard Schoen and Shing-Tung Yau

A fundamental nonlinear object in differential geometry is a map between manifolds. If the manifolds have Riemannian metrics, then it is natural to choose representaives for maps which respect the metric structures of the manifolds. Experience suggests that one should choose maps which are minima or critical points of variational integrals. Of the integrals which have been proposed, the energy has attracted most interest among analysts, geometers, and mathematical physicists. Its critical points, the harmonic maps, are of some geometric interest. They have also proved to be useful in applications to differential geometry. Particularly one should mention the important role they play in the classical minimal surface theory. Secondly, the applications to Kahler geometry given in [S], [SiY] illustrate the usefulness of harmonic maps as analytic tools in geometry. It seems to the author that there is good reason to be optimistic about the role which the techniques and results related to this problem can play in future developments in geometry.

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Abstract:This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions.Keywords: Bochner technique; Riemannian and Kähler manifolds; conformal diffeomorphism; Liouville-type theorem; scalar curvature; paracomplex structureMSC:53C20 2ff7e9595c