Nonorientable minimal surfaces with catenoidal ends
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Nonorientable minimal surfaces with catenoidal ends Kohei Hamada1 · Shin Kato1 Received: 19 June 2020 / Accepted: 12 October 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Starting from the pioneering work by Meeks, complete nonorientable minimal surfaces with finite total curvature have been studied by many researchers. However, it seems that there are no known examples all of whose ends are embedded except for Kusner’s flat-ended N-noids. In this paper, we show the existence of a 1-parameter family of complete 𝐙N -invariant conformal minimal immersions from finitely punctured real projective planes into 𝐑3 , each of which has N + 1 catenoidal ends, for any odd integer N ≥ 3 . This family gives a deformation from an (N + 1)-noid with N catenoidal ends and a planar end to Kusner’s flat-ended N-noid. We also give a nonexistence result for such surfaces for any even integer N ≥ 2. Keywords Minimal surface · Nonorientable · Flux formula · Catenoidal end Mathematics Subject Classification Primary 53C42 · Secondary 58E12
1 Introduction ̃ → M be a double covering map defined on Let M be a nonorientable surface, and 𝜋 ∶ M ̃ . Let X ∶ M → 𝐑3 be a complete conformal minimal immersion with a Riemann surface M ̃ is also ̃ ∶= X◦𝜋 ∶ M ̃ → 𝐑3 . Since X finite total curvature. Then, a natural lift is given by X ̃ a complete conformal minimal immersion with finite total curvature, by Osserman [9], M is conformally equivalent with a compact Riemann surface M ̃ punctured at a finite number ̃ extends holomorphically on M ̃ → 𝐒2 of X of points, and the Gauss map G ∶ M ̃ . We call each of these puncturing points, or the image of some neighborhood of the point, an end of ̃ . In the same way, we call the image of each of these points by 𝜋 , an end of X. Since 𝜋 is a X ̃ is even. double covering map, the number of the ends of X ̃ coincides with −4d𝜋 . ̃ of X Denote the degree of G by d. Then, the total curvature TC(X) Hence, the total curvature TC(X) of X is −2d𝜋 . Since catenoids and Enneper’s surfaces only
* Shin Kato [email protected]‑cu.ac.jp Kohei Hamada [email protected]‑cu.ac.jp 1
Department of Mathematics, Osaka City University, 3‑3‑138 Sugimoto, Sumiyoshi‑ku, Osaka 558‑8585, Japan
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have the total curvature −4𝜋 , d = 1 cannot be attained by any nonorientable one. Meeks [8, Theorem 2] gave the first example of a complete conformal minimal immersion defined on a real projective plane 𝐑P2 punctured at a point, that is a Möbius strip, with total curvature −2 ⋅ 3𝜋 = −6𝜋. In general, it is known as Chern–Osserman’s inequality that any complete conformal minin points satisfies mal immersion with finite total curvature defined on M ̃ punctured at ̃
̃ ̃ ≤ −4(̃ TC(X) n − 1 + genus(M))𝜋. This implies the inequality for complete conformal minimal immersions defined on M punctured at n points as follows:
̃ TC(X) ≤ −2(2n − 1 + genus(M))𝜋. In particular, for each inequality, the equality holds if and onl
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