## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 43, Issue 2 (1999), 222-232.

### On the image of the Gauss map of an immersed surface with constant mean curvature in $\mathbb{R}^{3}$

Nedir Do Espírito-Santo, Katia Frensel, and Jaime Ripoll

#### Abstract

We prove, generalizing a well known property of Delaunay surfaces, that if the Gauss image of a cmc surface in the Euclidean space is a compact surface with boundary, then any connected component of sphere minus the image is a strictly convex domain. We also obtain conditions under which the Gauss image has a regular boundary. These results relate to the question, raised by do Carmo, of whether the Gauss image of a complete cmc surface contains an equator of the sphere.

#### Article information

**Source**

Illinois J. Math., Volume 43, Issue 2 (1999), 222-232.

**Dates**

First available in Project Euclid: 19 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1255985211

**Digital Object Identifier**

doi:10.1215/ijm/1255985211

**Mathematical Reviews number (MathSciNet)**

MR1703184

**Zentralblatt MATH identifier**

0958.53009

**Subjects**

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

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

#### Citation

Do Espírito-Santo, Nedir; Frensel, Katia; Ripoll, Jaime. On the image of the Gauss map of an immersed surface with constant mean curvature in $\mathbb{R}^{3}$. Illinois J. Math. 43 (1999), no. 2, 222--232. doi:10.1215/ijm/1255985211. https://projecteuclid.org/euclid.ijm/1255985211