Download The Geometry of Complex Domains by Robert E. Greene, Kang-Tae Kim, Steven G Krantz PDF
By Robert E. Greene, Kang-Tae Kim, Steven G Krantz
The geometry of complicated domain names is a topic with roots extending again greater than a century, to the uniformization theorem of Poincaré and Koebe and the ensuing evidence of life of canonical metrics for hyperbolic Riemann surfaces. nowa days, advancements in numerous advanced variables by way of Bergman, Hörmander, Andreotti-Vesentini, Kohn, Fefferman, and others have unfolded new percentages for the unification of complicated functionality conception and complicated geometry. particularly, geometry can be utilized to review biholomorphic mappings in notable methods. This publication provides an entire photo of those developments.
Beginning with the one-variable case—background info which can't be stumbled on in other places in a single place—the ebook provides a whole photograph of the symmetries of domain names from the perspective of holomorphic mappings. It describes the entire proper options, from differential geometry to Lie teams to partial differential equations to harmonic research. particular options addressed include:
- covering areas and uniformization;
- Bergman geometry;
- automorphism groups;
- invariant metrics;
- the scaling method.
All sleek effects are followed by means of specified proofs, and plenty of illustrative examples and figures look throughout.
Written by way of 3 major specialists within the box, The Geometry of advanced Domains is the 1st ebook to supply systematic therapy of modern advancements within the topic of the geometry of advanced domain names and automorphism teams of domain names. a distinct and definitive paintings during this topic quarter, will probably be a priceless source for graduate scholars and an invaluable reference for researchers within the field.
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Extra info for The Geometry of Complex Domains
Example text
Thus the distance from ϕj (zj ) to ϕj (z0 ) is bounded by M zj − z0 , and hence goes to 0. 8 that {ϕj } has a subsequence that converges to some ϕ0 ∈ Aut (Ω). The compactness of {(ϕ, z) : (ϕ(z), z) ∈ C} has thus been established. 13. If Ω is a bounded domain in Cn , then Aut (Ω) is a Lie group. Proof. 11). 1, this result implies, from the result of Palais [Palais 1961], the existence of a smooth Riemannian metric on Ω invariant under Aut (Ω). Averaging this with respect to the almost complex structure produces a Hermitian metric on Ω invariant under Aut (Ω).
A familiar analysis shows that the possible Riemann surface quotients that can thus arise are C \ {0}, topologically a cylinder, and surfaces of genus 1, topologically tori. And these possibilities do indeed occur. But these are the only topological possibilities. All other Riemann surfaces must be quotients of the disc D. Clearly the possibilities for groups acting on D to yield Riemann surface covering quotients must be many and varied. But, as we shall see, much can be said about this situation in spite of its generality.
13. If Ω is a bounded domain in Cn , then Aut (Ω) is a Lie group. Proof. 11). 1, this result implies, from the result of Palais [Palais 1961], the existence of a smooth Riemannian metric on Ω invariant under Aut (Ω). Averaging this with respect to the almost complex structure produces a Hermitian metric on Ω invariant under Aut (Ω). In Chapter 3, an explicit construction of such a metric will be presented, but it is worth noting that the existence of such an invariant metric is guaranteed by the general principles we have discussed.