Robert E. Greene

Deformation of complex structures, estimates for the ∂ equation, and stability of the Bergman kernel

The purpose of this paper is to investigate the stability, under perturbation of the boundary or ,of the complex structure, of the solutions to the

The geometry of complex domains

Neumann problem on smoothly bounded strongly pseudoconvex domains and of the Fefferman asymptotic expansion of the Bergman kernel on these domains. The significance of these results arises in part from the fact that there is little hope of realizing the Bergman kernel explicitly except in the restricted case of homogeneous domains. Since a homogeneous C”O strongly pseudoconvex domain is necessarily biholomorphic to the ball, it is thus only through general results, such as the asymptotic expansion, that the Bergman kernels of strongly pseudoconvex domains not biholomorphic to the ball can be studied. The stability of the Bergman kernel has two aspects:. (i) Stability of behavior in the region where the kernel is C”O and bounded (i.e., pairs of points which are away from the boundary or from each other) and (ii) stability in the region consisting of pairs of points simultaneously near the boundary and near each other, where the kernel becomes unbounded as the boundary is approached. Specifically if D is a C” strongly pseudoconvex domain in C” and if E, = {(z, w) E b x 6: ] z w ( + dis(z, aD) + dis(w, 80) < S}, then the Bergman kernel function K,: D X D + Cc is CW on D x DIE, for any positive 6 [37]. Here it will be shown (Theorem 3.38) that for fixed 6 > 0, the Cw function KD ]orXWII varies continuously in the C” topology. This result is established using a stability result (Theorem 3.10) for the &Neumann operator which states in effect that P small perturbations of the complex structure of a fixed domain result in a small perturbation of the Neumann operator for the Cauchy-Riemann complex which is small in

An Interior Fixed Point Property of the Disc

Preface -- 1 Preliminaries -- 2 Riemann Surfaces and Covering Spaces -- 3 The Bergman Kernel and Metric -- 4 Applications of Bergman Geometry -- 5 Lie Groups Realized as Automorphism Groups -- 6 The Significance of Large Isotropy Groups -- 7 Some Other Invariant Metrics -- 8 Automorphism Groups and Classification of Reinhardt Domains -- 9 The Scaling Method, I -- 10 The Scaling Method, II -- 11 Afterword -- Bibliography -- Index.

Relative isometric embeddings of Riemannian manifolds

1. STIRRING THE COFFEE. At the end of this evenings meal, stir your cup of coffee and watch the motion of the top of the coffee. You will probably notice that there seem to be points where the coffee is not moving. The Brouwer Fixed Point Theorem, implies that such points must always exist. To be precise, suppose you could set the whole top surface of the coffee in motion at once. Then after a moment, every point of the surface would have moved. We could think of this motion as defining a map, that is a continuous function, f: D -> D of the closed disc (the top surface) to itself. But Brouwer proved that the disc has the fixed point property, that is, that every map from D to itself has a fixed point: a point p E D such that f(p) = p. This contradiction shows that there is a point on the surface where the coffee is not moving. You will probably observe some fixed points, but they may turn out to be difficult to detect. Since Brouwers Theorem says nothing about the location of the fixed points, the fixed points might lie only on the boundary of the disc. If the way you stir your coffee produces a map with fixed points of this kind, it may be hard to convince you that Brouwers Theorem is true because the entire (interior) surface of the coffee would be in motion, with fixed points only where the coffee touches the cup. If a map f: D --> D moves every point on the boundary of D, which we shall call S, then f must have a fixed point in int(D), the interior of D, since f must have a fixed point somewhere on D. But when f does have a fixed point on S, then f may or may not have additional fixed points in int(D). It is natural to ask what one can conclude about the interior fixed points just from knowing the restriction of f to the boundary S, which we denote by f IS. We will be concerned with the case where f IS takes S to itself. If f IS: S -> S has at least one fixed point, then there is a map G: D --> D with no interior fixed points such that GIS = f IS, as we will show in the next section. What is surprising is that, even if f IS: S --> S is smooth, that is, continuously differentiable, there may not be any smooth map G: D -> D with GIS = f IS that has no interior fixed points. In joint research with Helga Schirmer [1], we investigated when smooth extensions of maps on the boundary of a compact differentiable manifold must have interior fixed points. Our goal in this article is to explain these ideas in a concrete and easily-visualized setting, the 2-dimensional disc. Most of the things that can be done continuously in topology can be done smoothly also, because of general results about smooth approximation of continuous functions. But there are some exceptions, usually surprising, to that general principle. We will tell you about a new surprising exception.

Smooth realizations of relative Nielsen numbers

We prove the existence of C 1 isometric embeddings, and C ∞ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.

The Riemann mapping theorem from Riemann’s viewpoint

Abstract It is known that the relative Nielsen number N ( f ; X , A ), the Nielsen number of the complement N ( f ; X − A ) and the Nielsen number of the closure N(f; X − A ) are optimal lower bounds for the number of fixed points on X , X − A and X − A for self-maps of pairs of compact polyhedra ( X , A ) which satisfy fairly general assumptions. We show here that this is still true in the smooth category, i.e., that under equivalent assumptions these Nielsen type numbers are optimal lower bounds for the number of fixed points of smooth self-maps on pairs of smooth manifolds.

The Significance of Large Isotropy Groups

This article presents a rigorous proof of the Riemann mapping theorem via Riemann’s method, uncompromised by any appeals to topological intuition.

Automorphism Groups and Classification of Reinhardt Domains

Two facts about isotropy groups have played a central role up to now in our study of automorphism groups of bounded domains: First, that assigning to each element of the isotropy its differential at the fixed point gives an injective isomorphism onto a subgroup of the linear group of invertible linear maps of the tangent space at the point to itself (Corollary 1.3.3).

The Bergman Kernel and Metric

This chapter will give a brief survey of results about the automorphisms of domains that possess circular symmetries. They are a rich source of examples in the study of invariant geometry and automorphism groups.

Riemann Surfaces and Covering Spaces

The subject of this chapter is a remarkable construction developed by Stefan Bergman that produces an explicit, smooth, automorphism-invariant Hermitian metric on each bounded domain in \(\mathbb{C}{^n}\).