Logarithmically spiraling helicoids

Abstract

We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals. The surfaces have a self-similarity inherited from the curves and the nature of the construction. Moreover, inside of a "logarithmic cone", the surfaces are embedded.MSC 53A05, 53C21. Differential geometry and minimal surfaces and partial differential equations and perturbation methods 1

Explicit Björling Surfaces with Prescribed Geometry

“…Theorem 2.3. Given polyexp vectors e 1 (t), e 2 (t), e 3 (t) ∈ R 3 and a nonvanishing polyexp function µ(t) such that 1 µ (e 1 , e 2 , e 3 ) ∈ SO(3), we can explicitly find a curve c(t) with c (t) = e 1 (t). Then, the curve c(t) and the rotating normal…”

“…In several recent papers (eg [1,3,5,8,9]) embedded minimal disks have been constructed that have the appearance of a coil. They contain a core curve along which the surface normal rotates in a controllable way.…”