By Kichoon Yang
This monograph comprises an exposition of the speculation of minimum surfaces in Euclidean house, with an emphasis on whole minimum surfaces of finite overall curvature. Our exposition relies upon the philosophy that the research of finite overall curvature entire minimum surfaces in R3, in huge degree, coincides with the examine of meromorphic services and linear sequence on compact Riemann sur faces. This philosophy is first indicated within the primary theorem of Chern and Osserman: an entire minimum floor M immersed in R3 is of finite overall curvature if and provided that M with its caused conformal constitution is conformally similar to a compact Riemann floor Mg punctured at a finite set E of issues and the tangential Gauss map extends to a holomorphic map Mg _ P2. therefore a finite overall curvature whole minimum floor in R3 offers upward thrust to a airplane algebraic curve. allow Mg denote a set yet differently arbitrary compact Riemann floor of genus g. a favorable integer r is named a puncture quantity for Mg if Mg should be conformally immersed into R3 as an entire finite overall curvature minimum floor with precisely r punctures; the set of all puncture numbers for Mg is denoted via P (M ). for instance, Jorge and Meeks [JM] confirmed, via developing an instance g for every r, that each optimistic integer r is a puncture quantity for the Riemann floor pl.
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Extra info for Complete Minimal Surfaces of Finite Total Curvature
As a consequence, we have a well-defined complex structure on the two-manifold (M, ds 2 ). We state this as a theorem. Theorem 29 On an oriented Riemannian two-manifold (M, ds 2 ) the totality of oriented isothermal coordinate systems (x, V) defines a complex structure by letting the z's, z = x + iV, be holomorphic coordinates. Hereafter, an oriented Riemannian two-manifold (M, ds 2 ) will always be thought of as a Riemann surface as well via the induced complex structure described in the above theorem.
Consider the map Q : f Rd (M) -+ Div! (M) -+ J (M), 1--+ D = (f)oo 1--+ u (D). An easily verified fact is that for any f E Rd (M) and A E Aut (PI) the divisors (f)oo and (A 0 f)oo are linearly equivalent to each other (see below for a proof). From this we surmise that the map Q descends to a map Note also that x E J (M) is in the image of Q if and only if the complete linear series u- I (x) is base~point-free by an earlier proposition. We are interested in examining the set Q (Rd (M)) n (Wd - Wd+I) C J (M) , r ~ 1.
6) We shall see shortly that this map is, in fact, an isomorphism; we will use the identification Div (M) /linear equivalence = HI (M, 0*), [D] = L D . The holomorphic sections of a line bundle L _ M is denoted by HO (L), and we will let hO (L) denote its dimension. Given a line bundle its degree can be computed by the integral 1M CI (L), CI (L) = the Chern class of L. Another method, which is more useful in our setting, of finding the degree of a line bundle is to take the degree of any divisor representing it.