By Dror Varolin
This publication establishes the elemental functionality thought and intricate geometry of Riemann surfaces, either open and compact. a few of the equipment utilized in the publication are diversifications and simplifications of equipment from the theories of numerous complicated variables and intricate analytic geometry and might function very good education for mathematicians desirous to paintings in complicated analytic geometry. After 3 introductory chapters, the ebook embarks on its vital, and definitely so much novel, objective of learning Hermitian holomorphic line bundles and their sections. between different issues, finite-dimensionality of areas of sections of holomorphic line bundles of compact Riemann surfaces and the triviality of holomorphic line bundles over Riemann surfaces are proved, with quite a few functions. probably the most results of the ebook is Hörmander's Theorem at the square-integrable answer of the Cauchy-Riemann equations. The crowning program is the evidence of the Kodaira and Narasimhan Embedding Theorems for compact and open Riemann surfaces. The meant reader has had first classes in genuine and complicated research, in addition to complex calculus and uncomplicated differential topology (though the latter topic isn't crucial). As such, the e-book may still entice a wide component to the mathematical and clinical neighborhood. This e-book is the 1st to provide a textbook exposition of Riemann floor concept from the perspective of optimistic Hermitian line bundles and Hörmander $\bar \partial$ estimates. it truly is extra analytical and PDE orientated than past texts within the box, and is a superb creation to the equipment used at present in complicated geometry, as exemplified in J. P. Demailly's on-line yet differently unpublished e-book "Complex analytic and differential geometry." I used it for a one area direction on Riemann surfaces and located it to be essentially written and self-contained. It not just fills an important hole within the huge textbook literature on Riemann surfaces yet can also be fairly indispensible in the event you wish to train the topic from a differential geometric and PDE perspective.
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These notions are invariant under local biholomorphic functions, and thus can be extended to Riemann surfaces. 4. Let f be holomorphic in a punctured neighborhood of a point p ∈ X (a) We say that f has a removable singularity at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has a removable singularity at ϕ(p). 50 (b) We say that f has a pole of order n at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has a pole of order n at ϕ(p). (c) We say that f has an essential singularity at p if there exists a complex chart ϕ : U → V containing p such that f ◦ ϕ−1 has an essential singularity at ϕ(p).
4 Prove that if f is holomorphic on the punctured unit disk D − {0} and then f ∈ O(D). 5 Let F ⊂ O(C) be a family of entire functions such for each R > 0 there is a constant CR such that |fj |2 dA ≤ CR . sup f ∈F D(0,R) Show that every sequence in F has a convergent subsequence. 6 Find a harmonic function in the punctured unit disk that is not the real part of a holomorphic function. 7 Let u be a subharmonic function in the unit disk. Show that the function ψ : [0, ε) → R ∪ [−∞) defined by 2π √ 1 ψ(r) := u(er e −1θ )dθ 2π 0 is convex and increasing.
17. If f ∈ M (X) is a meromorphic function with a single, simple pole, then X is isomorphic to P1 . Proof. By hypothesis, Deg(F ) = #(F −1 (∞)) = 1, and thus by the previous corollary, F is an isomorphism. R EMARK . We can also give another proof that the sum of the orders of a meromorphic function on a compact Riemann surface is 0. Indeed, according to the theorem on degree, Deg(f ) = Multx (f ) = f (x)=0 Multy (f ). 13. 18. Let S be a compact 2-manifold, possibly with (smooth) boundary. 1. A 0-simplex, or vertex, is a point, a 1-simplex, or edge, is a set homeomorphic to a closed interval, and a 2-simplex, or face, is a set homeomorphic to the triangle {(x, y) ∈ [0, 1] × [0, 1] ; x + y ≤ 1}.