Bounds on the topology and index of minimal surfaces

Abstract

We prove that for every nonnegative integer g, there exists a bound on the number of ends of a complete, embedded minimal surface M in R 3 of genus g and finite topology. This bound on the finite number of ends when M has at least two ends implies that M has finite stability index which is bounded by a constant that only depends on its genus.Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

The local picture theorem on the scale of topology

“…By the regular neighborhood theorem in [37] or [46], the surfaces in B a,b all have cubical area growth, i.e., Remark 5.4 With the notation of Theorem 1.1 in this paper, if M has finite genus or if the sequence {λ n M n } n has uniformly bounded genus in fixed size intrinsic metric balls, then item 6 of that theorem does not occur, since item (B) of Proposition 4.30 does not occur. This fact will play a crucial role in our forthcoming paper [25], when proving a bound on the number of ends for a complete, embedded minimal surface of finite topology in R 3 , that only depends on its genus. Also in [27], we will apply Theorem 1.1 to give a general structure theorem for singular minimal laminations of R 3 with a countable number of singularities.…”

“…Proof of Proposition 4.23. As the minimal surface Σ(t 2 ) defined in (25) can be considered to be a proper subdomain of the surface Σ(t 2 ) defined immediately before (26), and Σ(t 2 ) is conformally diffeomorphic to a closed halfplane (by Lemmas 4.28 and 4.29), then Σ(t 2 ) is a parabolic surface. The restriction of the x 3 -coordinate function to Σ(t 2 ) is a bounded harmonic function with boundary values greater than or equal to m = min{x 3 (q) | q ∈ α γ(t 2 ) } > 0.…”

“…Section 5 includes some applications of Theorem 1.1. We refer the reader to [23,24,25,26,27,36,38,40] for further applications of Theorem 1.1.…”

Structure theorems for singular minimal laminations

“…(II) In [21] we will apply Theorem 1.4 to prove that for each g ∈ N ∪ {0}, there exists a bound on the number of ends of a complete, embedded minimal surface in R 3 with finite topology and genus at most g. This topological boundedness result implies that the stability index of a complete, embedded minimal surface of finite index in R 3 has an upper bound that depends only on its finite genus.…”

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We apply the local removable singularity theorem for minimal laminations [31] and the local picture theorem on the scale of topology [23] to obtain two descriptive results for certain possibly singular minimal laminations of R 3 . These two global structure theorems will be applied in [21] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in R 3 , and in [22] to prove that a complete, embedded minimal surface in R 3 with finite genus and a countable number of ends is proper.

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