Doubly connected minimal surfaces and extremal harmonic mappings

Details Doubly connected minimal surfaces and extremal harmonic mappings Doubly connected minimal surfaces and extremal harmonic mappings

2009-12-17

arxiv.org/abs/0912.3542v2

Mathematics

Differential Geometry

35 pages, 2 figures. Minor edits, references added

Scientific paper

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.

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