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• W2019921193 abstract "The classical Henneberg's minimal surface (1875, [3, 4, 11]) was the unique nonorientable example known until 1981, when Meeks [6] exhibited the first example of a nonorientable, regular, complete, minimal surface of finite total curvature -6ir. In this paper, we study the nonorientable, regular, complete minimal surfaces of finite total curvature and give some examples of punctured projective planes regularly and minimally immersed in R3 and Rn. 1. Nonorientable minimal surfaces in Rn. We consider surfaces S in Rn defined by maps X: M -+ R , where M is a two-dimensional manifold. For the study of a nonorientable, connected surface S we take the double surface S given by X: M -? R , where II: M-? M is the oriented two-sheeted covering of M, and X = X o I. We have an involution I: M -M without fixed points and a conformal structure on M such that I is antiholomorphic. We have a representation theorem for the nonorientable minimal strfaces: THEOREM 1. 1. Let S be a nonorientable regular connected minimal surface in Rn. The double surface S is a minimal surface in Rn defined by X: M -Rn, (1.1) X(p) = Re f0(S)dk, PoNP E M, with a = q(j)d' such that I*a = a (that is, I*ak = ?k, 1 < k < n). Conversely, if S is a regular orientable connected minimal surface in Rn given by (1.1) and if there is an antiholomorphic involution I: M -? M without fixed points such that i* a = a, then S is the double surface of a regular nonorientable minimal surface in Rn. The consequences of the condition I*ca = a with respect to the various forms of representation for minimal surfaces in Rn are COROLLARY 1.2 (MEEKS [6]). Let f and g be the functions of Weierstrass 's representation of an orientable, regular, connected minimal surface S in R3, that X(p) =Re 2 g2(f), i(l + g2(s)), 29(f)) d?, polp EM The surface S is the double surface of a nonorientable, regular minimal surface in R3 if and only if 1.2) J(i) g(I(p)) = -l/g(p), p E M, and (1.2) 1 (ii) I*(w) = -92w, W = f(z) dz, for some antiholomorphic involution I: M M without fixed points. Received by the editors October 31, 1985. 1980 Mathiernatic Subect Cakssification (1985 Revtiion). Primary 53A10; Secondary 30A68. (?)1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 629 This content downloaded from 157.55.39.185 on Thu, 26 May 2016 06:29:25 UTC All use subject to http://about.jstor.org/terms 630 M. E. G. G. DE OLIVEIRA COROLLARY 1.3. Let S be an orientable, regular, connected minimal surface in R4 given by :M R4 X(p) = Re (1 + 9192, i(192), 91 92, -i(91 + 92))W, po,p E M. Then, S is the double surface of a nonorientable minimal surface in R4 if and only if, for some antiholomorphic involution I on M without fixed points, 1.3) t (i) gk(I(P)) = -1/gk(p), k = 1,2, and (1.3) l(ii) I*~w = 9192Wi w = f(z) dz, If C(S) is the total curvature of S, we can define the total.curvature of S by (1.4) C(S) = C(S)/2. In what follows, the double surface S will always be connected, complete and of finite total curvature C(S) = 27rfi2. From Chern and Osserman's theorem [1] we have M conformally equivalent to a compact Riemann surface of genus j punctured at i = 2r points {P1,P2, .. iPr,ql .. Iqr }, with I(pj) = qj, I(pj) being the extension of I to pj, 1 < j < r. The function X of (1.1) has poles of order m3 at pj and qj, and r (1.5) ff 2 E mj = 2-y 2. j=j The Chern and Osserman inequality for nonorientable minimal surfaces is (1.6) C(S) < 27r(X(M) r) with x(M) the Euler characteristic of M. A nonorientable version of Gackstatter's theorem [2] gives an estimate for the dimension of a nonorientable, complete, minimal surface S with total curvature C(S) = -27rm, r ends and genus -y: (1.7) Dim(S) < 2m 2-y r + 3." @default.
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• W2019921193 date "1986-04-01" @default.
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• W2019921193 title "Some new examples of nonorientable minimal surfaces" @default.
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