## Illinois Journal of Mathematics

### A class of surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ associated to harmonic functions and a relation between CMC-1/2 and flat surfaces

#### Abstract

We introduce a geometric motivated method to construct immersions into $\mathbb{H}^2\times\mathbb{R}$ from a smooth function $\varphi$ defined on an open set of the unit disc, and study the relation between the geometry of the immersion in terms of partial differential equations for $\varphi$. We give two applications of the method. First, we introduce the class of surfaces generated by harmonic functions and show that they have properties analogous to minimal surfaces in $\mathbb {R}^3$. We also exhibit an explicit local relation between CMC 1/2 and flat surfaces in $\mathbb{H}^2\times\mathbb{R}$.

#### Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1123-1145.

Dates
First available in Project Euclid: 18 November 2009

https://projecteuclid.org/euclid.ijm/1258554353

Digital Object Identifier
doi:10.1215/ijm/1258554353

Mathematical Reviews number (MathSciNet)
MR2595758

Zentralblatt MATH identifier
1181.53009

#### Citation

Ferreira, Walterson; Roitman, Pedro. A class of surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ associated to harmonic functions and a relation between CMC-1/2 and flat surfaces. Illinois J. Math. 52 (2008), no. 4, 1123--1145. doi:10.1215/ijm/1258554353. https://projecteuclid.org/euclid.ijm/1258554353

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