## Acta Mathematica

### Helicoidal minimal surfaces of prescribed genus

#### Abstract

For every genus g, we prove that ${\mathbf{S}^2\times\mathbf{R}}$ 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 ${\mathbf{S}^2}$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in ${\mathbf{R}^3}$ that are helicoidal at infinity. We prove that helicoidal surfaces in ${\mathbf{R}^3}$ of every prescribed genus occur as such limits of examples in ${\mathbf{S}^2\times\mathbf{R}}$.

#### Note

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

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

Dates
Revised: 25 July 2015
First available in Project Euclid: 30 January 2017

https://projecteuclid.org/euclid.acta/1485802480

Digital Object Identifier
doi:10.1007/s11511-016-0139-z

Mathematical Reviews number (MathSciNet)
MR3573331

Zentralblatt MATH identifier
1356.53010

Rights

#### Citation

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. https://projecteuclid.org/euclid.acta/1485802480

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