Communications in Analysis and Geometry

Volume 26 (2018)

Number 3

Logarithmically spiraling helicoids

Pages: 461 – 504



Christine Breiner (Department of Mathematics, Fordham University, Bronx, New York, U.S.A.)

Stephen J. Kleene (Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)


We construct helicoid-like embedded minimal disks with axes along self-similar curves modeled on logarithmic spirals using PDE methods. 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.


differential geometry, minimal surfaces, partial differential equations, perturbation methods

2010 Mathematics Subject Classification

53A05, 53C21

C. Breiner was supported in part by NSF grant DMS-1308420 and by an AMS-Simons Travel Grant. S. J. Kleene was partially supported by NSF grant DMS-1004646.

Received 10 June 2014

Published 27 July 2018