Variation of hodge structure: The singularities of the period mapping

The subject of these lecture notes is comparison theory in Riemannian geometry: What can be said about a complete Riemannian manifold when (mainly lower) bounds for the sectional or Ricci curvature are given? Starting from the comparison theory for the Riccati ODE which describes the evolution of the principal curvatures of equidistant hypersurfaces, we discuss the global estimates for volume and length given by Bishop-Gromov and Toponogov. An application is Gromov’s estimate of the number of generators of the fundamental group and the Betti numbers when lower curvature bounds are given. Using convexity arguments, we prove the ”soul theorem” of Cheeger and Gromoll and the sphere theorem of Berger and Klingenberg for nonnegative curvature. If lower Ricci curvature bounds are given we exploit subharmonicity instead of convexity and show the rigidity theorems of Myers-Cheng and the splitting theorem of Cheeger and Gromoll. The Bishop-Gromov inequality shows polynomial growth of finitely generated subgroups of the fundamental group of a space with nonnegative Ricci curvature (Milnor). We also discuss briefly Bochner’s method. The leading principle of the whole exposition is the use of convexity methods. Five ideas make these methods work: The comparison theory for the Riccati ODE, which probably goes back to L.Green [15] and which was used more systematically by Gromov [20], the triangle inequality for the Riemannian distance, the method of support function by Greene and Wu [16],[17],[34], the maximum principle of E.Hopf, generalized by E.Calabi [23], [4], and the idea of critical points of the distance function which was first used by Grove and Shiohama [21]. We have tried to present the ideas completely without being too technical. These notes are based on a course which I gave at the University of Trento in March 1994. It is a pleasure to thank Elisabetta Ossanna and Stefano Bonaccorsi who have worked out and typed part of these lectures. We also thank Evi Samiou and Robert Bock for many valuable corrections.

Comparison Theorems in Riemannian Geometry

(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

On etudie le comportement asymptotique de varietes de Riemann. On construit des coordonnees a l'infini sous certaines conditions intrinseques

On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth

The complex analysis of strictly pseudoconvex domains in IE" is rather well known, especially since the boundary regularity properties of solutions of the inhomogeneous Cauchy-Riemann equations 8c~=fl on such domains have been described more and more precisely and important consequences from this have been derived (for a survey of the results in this direction known until 1972 (see [8]), more recent results are for instance contained in [15, 23]). In contrast to this situation, many fundamental questions with respect to the most "natural" domains of complex analysis, namely arbitrary pseudoconvex domains in ~ ' , are still open. What is the reason for this discrepancy? One possible answer to this question can be seen in the following observation. In almost all important proofs concerning the analysis of strictly pseudoconvex domains, at least one out of three elementary properties of these domains play a fundamental role. They are: 1) Strict pseudoconvexity is stable with respect to small C2-perturbations. 2) A strictly pseudoconvex domain is locally biholomorphically equivalent to strictly convex domains. 3) If ( 2 c c ~ " is strictly pseudoconvex with a smooth C2-boundary, then there is a neighborhood U of ~ and a strictly plurisubharmonic C 2 function p on U such that dp(p):t=O for peb~2 and

Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions

/pdf/normal-families-of-holomorphic-mappings-2zrcv7g89l.pdf

Normal families of holomorphic mappings

/pdf/c-convex-functions-and-manifolds-of-positive-curvature-1bs78x7tme.pdf

C∞ convex functions and manifolds of positive curvature

This is the first of two papers devoted to the study of compact Kghler manifolds of nonnegative bisectional curvature (see also [Wu2]). The main result of this paper is the following theorem; together with its corollaries below, this theorem shows that such manifolds possess a rigid internal structure. For their statements, recall from [Wul] that the Rieei curvature is quasi-positive iff it is everywhere nonnegative and is positive in all directions at a point; an equivalent definition is that the Ricci tensor Ric is everywhere positive semi-definite and is positive definite at a point.

/pdf/on-compact-kahler-manifolds-of-nonnegative-bisectional-byl9fd62fp.pdf

On compact Kähler manifolds of nonnegative bisectional curvature, II

A standard technique in classical analysis for the study of eontinous sub-solutions of the Dirichlet problem for second order operators may be illustrated as follows. Suppose it is to be shown that a continuous real function ](x) is convex (respectively, striely convex) at x0; then it suffices to produce a C ~ function g(x) such that g(x)<<.](x) near x 0 and g(Xo) =/(x0), and such that 9\"(xo) >/0 (respectively g\"(xo) >1 some fixed positive constant). The main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone. Now in global differential geometry, the functions that naturally arise are often continuous but not differentiable. Since much of geometric analysis reduces to second order elliptic problems, this technique then recommends itself as a natural tool for overcoming this difficulty with the lack of differentiability. In a limited way, this technique has indeed appeared in several papers in complex geometry (e.g. Ahlfors [1], Takeuchi [20], Elenewajg [7] and Greene-Wu [11]; cf. also Suzuki [19]). The main purpose of this paper is to broaden and deepen the scope of this method by making it the central point of a general study of nonnegative sectional, Ricei or bisectional curvature. The following are the principal theorems; the relevant definitions can be found in Section 1. Let M be a noncompact complete Riemannian manifold and let 0 E M be fixed. Let {Ct}tG1 be a family of closed subsets of M indexed by a subset I of R. Assume that et = d(0, C t ) ~ as t ~ , where d(p, q) will always denote the distance between p, qEM relative to the Riemannian metric. The family of functions ~t: M-~R defined by ~t(P)=

/pdf/an-elementary-method-in-the-study-of-nonnegative-curvature-2x8ujf5m5w.pdf

An elementary method in the study of nonnegative curvature

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Papers

Normal families of holomorphic mappings

C∞ convex functions and manifolds of positive curvature

On compact Kähler manifolds of nonnegative bisectional curvature, II

An elementary method in the study of nonnegative curvature