Acta Mathematica

Helicoidal minimal surfaces of prescribed genus

David Hoffman, Martin Traizet, and Brian White

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For every genus g, we prove that S2×R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in R3 that are helicoidal at infinity. We prove that helicoidal surfaces in R3 of every prescribed genus occur as such limits of examples in S2×R.


The research of the second author was partially supported by ANR-11-ISO1-0002. The research of the third author was supported by NSF grants DMS–1105330 and DMS 1404282.

Article information

Acta Math., Volume 216, Number 2 (2016), 217-323.

Received: 20 June 2013
Revised: 25 July 2015
First available in Project Euclid: 30 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary: 53A10
Secondary: Secondary: 49Q05 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

2016 © Institut Mittag-Leffler


Hoffman, David; Traizet, Martin; White, Brian. Helicoidal minimal surfaces of prescribed genus. Acta Math. 216 (2016), no. 2, 217--323. doi:10.1007/s11511-016-0139-z.

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