4 research outputs found
Doubly connected minimal surfaces and extremal harmonic mappings
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.Comment: 35 pages, 2 figures. Minor edits, references adde