By Alfred Barnard Basset
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Additional info for A Treatise on the Geometry of Surfaces
C. Hyperbolic manifolds: A complete simply connected Kahler manifold with bisectional curvature less than a negative constant is biholomorphic to a proper subvariety of a bounded domain in Cn. Very little is known about this problem. We do not even know whether such a manifold is noncompact or not. Thus we shall restrict our attention to treat a simpler problem first. Namely we replace "bisectional curvature" 48 SHING-TUNG YAU by "sectional curvature" in the above problem. In this case, we know the geometry of the manifold quite well.
It will be interesting to show that any real analytic complete Kahler manifold with curvature bounded between zero and a negative constant can be compactified complex analytically if its volume is finite. If the manifold is locally symmetric, this was done by Satake [St] and Borel-Baily [BB]. §10. Harmonic maps The second class of quasilinear equation is the equation for harmonic mapping. It was first introduced and studied by Bochner [Boll as a generalization of minimal surfaces. They are interesting primarily because they tell us some geometric properties of mappings between Riemannian manifolds.
A compact surface is said to be infinitesimal rigid of order n if there is no nonzero deformation vector field of order n which vanishes on the boundary of the surface (if the boundary exists). It is easy to see that a piece of the plane is infinitesimal rigid of order two but not infinitesimal rigid of order one. The first major theorem proved in the theory of infinitesimal isometric deformation was due to Blaschke [B1]. It says that a closed surface whose curvature is nonnegative and is not equal to zero on any region is infini- tesimal rigid of order one.