Embedded minimal ends of finite type

Hauswirth, Laurent; Pérez, Joaquín; Romon, Pascal

doi:10.1090/S0002-9947-00-02640-4

Embedded minimal ends of finite type
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by Laurent Hauswirth, Joaquín Pérez and Pascal Romon PDF

Trans. Amer. Math. Soc. 353 (2001), 1335-1370 Request permission

Abstract:

We prove that the end of a complete embedded minimal surface in $\mathbb {R}^3$ 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 $\mathbb {R}^3$.

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