## New PDF release: The equidistribution theory of holomorphic curves

By Hung-his Wu

ISBN-10: 0691080739

ISBN-13: 9780691080734

This paintings is a clean presentation of the Ahlfors-Weyl thought of holomorphic curves that takes into consideration a few fresh advancements in Nevanlinna concept and a number of other complicated variables. The therapy is differential geometric all through, and assumes no earlier acquaintance with the classical thought of Nevanlinna. the most emphasis is on holomorphic curves outlined over Riemann surfaces, which admit a harmonic exhaustion, and the most theorems of the topic are proved for such surfaces. the writer discusses a number of instructions for additional research.

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**Additional resources for The equidistribution theory of holomorphic curves**

**Example text**

R-r g two form i s us fix an dz 1\ d-z. ~ a So they are trivially equal). We wish to construct a function S. € while its associated as so that ua(p) - 00 S - (a) equals ill. p - a 1 on ua c S on 2' dd u a and such that Let Such a function was first found by Ahlfors and Shimizu in 1929, and can be described in the following way. (ao'~) Let be a representative of e~e. mined up to a multiple of so quotient space of a in S (-81 '&0)' a; (Recall that unchanged if

And X(M') > X(M), M, f Consequently, if 0 ~ there exists no nonconstant holomorphic X(M) THE EQUIDlSTRIBlJflON THEORY OF HOLOMORPHIC CURVES 32 mappings f: M' - M. We shall generalize Hurwitz's Formula in Chapter IV; these are the so-called Pldcker formulas. §4. So far the open Riemann surface V has been arbitrary. To get deeper results, we must impose a restriction on V. 1. s.. , inverse image of compact set is compact. s.. reT) < s, reT), harmonic on such that T is T-l([r(T),s)). As usual, a fUnction T is harmonic iff respect to a local coordinate fUnction (l1T)dxl\dy, where l1 z ddcT =x + = ~ O.

As usual, let be the coordinate function on ~ 0 x becomes a holomorphic function on x*: U_c: 2 - (oJ map U be an such that x* Uo. :. :. reduced representation in any suffi- ciently small open set of §3. Thus x* enjoys the property that x x. clearly V. From this point on, let a fixed nonconstant holomor- x: V -+ S be given and let V with COO boundary. D be a compact subdomain In this section, ~ allow arbitrary, so in particular, it can be compact. we are interested in the counting multiplicity. n~ber Now of pre-images of x(p) = a iff V to be If a a in

### The equidistribution theory of holomorphic curves by Hung-his Wu

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