Complete Minimal Surfaces of Finite Total Curvature by Kichoon Yang

By Kichoon Yang

This monograph includes an exposition of the speculation of minimum surfaces in Euclidean area, with an emphasis on entire minimum surfaces of finite overall curvature. Our exposition is predicated upon the philosophy that the research of finite overall curvature whole minimum surfaces in R3, in huge degree, coincides with the learn of meromorphic capabilities and linear sequence on compact Riemann sur­ faces. This philosophy is first indicated within the basic 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 triggered conformal constitution is conformally reminiscent of 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 entire minimum floor in R3 supplies upward push to a airplane algebraic curve. allow Mg denote a hard and fast yet another way arbitrary compact Riemann floor of genus g. a favorable integer r is termed a puncture quantity for Mg if Mg could 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 through P (M ). for instance, Jorge and Meeks [JM] confirmed, through 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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If isothermal coordinates and if w = x + iii, then from the relation h (z) dz . dz = ds 2 = it (w) (x, y) are another dw . diiJ we see that dw = A . dz for some nonvanishing function A, or equivalently, ~~ 1= O. 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.

J (M) is surjective when the degree 38 Background Material Consider the Jacobi map on M: u: M ~ J (M), p 1-+ (i: (1, ... , i: (g) (mod periods), where Po E M is a base point and ((i) is a fixed basis for HO (M, K). Let us assume that 9 > 0, as J (PI) is just a point. Let z be a local coordinate centered at p, and write (i = ",idz. , p would have to be a base point of the canonical series. This shows that u is immersive on M. In fact, u : M ~ J (M) is an embedding as the following simple consideration shows: Suppose u (p) = u (q) = for some p #- q E M.

This shows that L has a nontrivial meromorphic section ~; hence, L is associated to a divisor, namely, the divisor of ~. As we saw in Section 3 every divisor gives rise to a complex vector space of meromorphic functions: L (D) = {! E H O (M, M *) : (/) 2: - D } U {O} . A nonzero element L (D) is a meromorphic function whose zero divisor is at least D- and whose polar divisor is at most D+. Fix a meromorphic section So of the line bundle LD = [D] - M. This gives rise to an isomorphism I] I]~-. ) For example, when D is integral we can think of HO ([D]) as the set of meromorphic functions on M whose polar divisors are at most D.

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