Embedded minimal ends of finite type

Abstract : We prove that the end of a complete embedded minimal surface in R^3 with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end E provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is C^0-asymptotic to a Helicoid. We apply these results to proving that any complete embedded one-ended minimal surface of finite type and infinite total curvature is asymptotic to a Helicoid, and we characterize the Helicoid as the only simply connected complete embedded minimal surface of finite type in R ^3.
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https://hal.archives-ouvertes.fr/hal-00175489
Contributor : Pascal Romon <>
Submitted on : Friday, September 28, 2007 - 1:31:12 PM
Last modification on : Wednesday, September 4, 2019 - 1:52:03 PM

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Laurent Hauswirth, Joaquín Pérez, Pascal Romon. Embedded minimal ends of finite type. Transactions of the American Mathematical Society, American Mathematical Society, 2000, 353 (4), pp.1335--1370. ⟨10.1090/S0002-9947-00-02640-4⟩. ⟨hal-00175489⟩

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