Abstract
In this paper we describe the notion of an annular end of a Riemann surface being of finite type with respect to some harmonic function and prove some theoretical results relating the conformal structure of such an annular end to the level sets of the harmonic function. As an application of these results, we obtain important information on the conformal type of any properly immersed minimal surface M in \(\mathbb {R}^3\) with compact boundary and which intersects some plane transversely in a finite number of arcs; in particular, such an M is a parabolic Riemann surface. This information is applied by Meeks III and Pérez (Embedded minimal surfaces of finite topology, 2015) to classify the asymptotic behavior of annular ends of a complete embedded minimal annulus with compact boundary in terms of the flux vector along its boundary. In the present paper, we apply this information to understand and characterize properly immersed minimal surfaces in \(\mathbb {R}^3\) of finite total curvature, in terms of their intersections with two nonparallel planes.
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Notes
A proper subdomain \(E\subset M\) is an annular end if it is homeomorphic to \(\mathbb {S}^1\times [0,1)\).
A noncompact Riemann surface \(\Sigma \) with boundary is hyperbolic if its boundary fails to have full harmonic measure (equivalently, bounded harmonic functions on \(\Sigma \) are not determined by their boundary values); otherwise, \(\Sigma \) is called parabolic.
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J. Pérez: Research supported in part by the MINECO/FEDER Grant no. MTM2014-52368-P.
W. H. Meeks III: This material is based upon work for the NSF under Award No. DMS-1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.
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Meeks, W.H., Pérez, J. Finite type annular ends for harmonic functions. Math. Ann. 367, 1047–1056 (2017). https://doi.org/10.1007/s00208-016-1407-0
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DOI: https://doi.org/10.1007/s00208-016-1407-0