Interpolation and optimal hitting for complete minimal surfaces with finite total curvature

2017/12/13 by Alarcon, Antonio, Castro-Infantes, Ildefonso, Lopez, Francisco J.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.04727

Abstract

We prove that, given a compact Riemann surface Σ and disjoint finite sets \varnothing≠ E⊂Σ and Λ⊂Σ, every map Λ→ ℝ³ extends to a complete conformal minimal immersion Σ∖ E→ ℝ³ with finite total curvature. This result opens the door to study optimal hitting problems in the framework of complete minimal surfaces in ℝ³ with finite total curvature. To this respect we provide, for each integer r≥ 1, a set A⊂ℝ³ consisting of 12r+3 points in an affine plane such that if A is contained in a complete nonflat orientable immersed minimal surface X\colon M→ℝ³, then the absolute value of the total curvature of X is greater than 4πr.

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• Uniform Approximation by Complete Minimal Surfaces of Finite Total Curvature in ℝ³
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