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

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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