Download Complete and Compact Minimal Surfaces by Kichoon Yang PDF

By Kichoon Yang

Those notes include an exposition of the idea of minimum surfaces with out boundary. There have been many fascinating contemporary advancements in the learn of minimum surfaces in quite a few Riemannian manifolds and soment a few of those fabrics offered from a constant point of view. major instruments used are the strategy of relocating frames and the conception of compact Riemann surfaces.

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Proposition. Let f: M ..... 1R3 be a conformal minimal immersion and also let {p" tp} be its Weierstrass representative. Then (30) where tp 11": = 11"0~~, S2 C 1R3 ..... ( U {co} is the stereographic projection and ~~: M ..... , ~~(z) = the unit normal vector perpendicular to f*(Tz(M)). §4. Algebraic Gauss Maps Digression. U(n) acts transitively on (pn-I as follows: Take v E (n\ {O} and let [v] E (pn-I denote the complex line through 0 and v. [v] = [Av]. The isotropy subgroup at t[l, 0, ... , 0] is U(l) x U(n-l) and (pn-I becomes a homogeneous space U(n)jU(I)xU(n-I).

V) Let M = (, cp(z) = z + ~, p,(z) = z2dz. Then the Gauss map of the resulting complete minimal surface is surjective. The following question is open: does there exist a complete conformal minimal immersion M --+ 1R3 with finite total curvature whose Gauss map misses three points? Chern [Cher2] (also Osserman [04]) gave a Bernstein type result for complete minimal surfaces in IRn : conformal minimal immersion. Let f. M --+ IRn be a complete nonplanar Then the subset of (,PI-I* (the space of hyperplanes in (,PI-I) meeting with ~(M) c (,PI-I is dense.

Also ds 2 = e*((n~)2 + ... + (n~)2) = (w1)2 + (w2)2 and (wI, w2) form a (local) orthonormal coframe on M. Index Convention. 0 5 a,/3,1 5 n, 1 5 i,j,k 5 2, 3 5 a,b,c 5 n.

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