XXVI International Fall Workshop on Geometry and Physics

Braga, 4-7 September 2017

Minimal surfaces ($ H=0$) in Euclidean 3-space $ \mathbb{R}^3 $ and Bryant surfaces ($ H=1$) in Hyperbolic 3-space $\mathbb{H}^3$ are a special family among all the constant mean curvature (CMC) surfaces in spaces forms. For example, they have a holomorphic Gauss map , which is a fundamental tool in the study of these surfaces. Similarly, surfaces of critical CMC in homogenous 3-spaces present a special behavior among all the CMC surfaces. As an example, in [FM] we defined a hyperbolic Gauss map for surfaces in $\mathbb{H}^2\times\mathbb{R}$ that turns out to be harmonic for surfaces of critical CMC. This Gauss map has been crucial in the proof of several results in the theory, as for example the resolution of the Bernstein problem in homogeneous spaces in [FM2]. Short after that, it was discovered in [Dan] a Gauss map for surfaces in Heisenberg space that is also harmonic when the mean curvature is critical. However, both definitions are quite different and it was unclear how to extend them to the general setting.
In this talk we will review some recent results on critical CMC surfaces in homogeneous 3-spaces and present the unified definition of a Gauss map for surfaces in these ambient spaces that is harmonic when the mean curvature of the surface is critical ([DFM]).

[Dan] B. Daniel. The Gauss map of minimal surfaces in the Heisenberg group. Int. Math. Res. Not. IMRN, 3 (2011), pp. 674-695.
[FM] I. Fernández and P. Mira. Harmonic maps and constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$. Amer. J. Math., 129 4 (2007), pp. 1145-1181.
[FM2] I. Fernández and P. Mira. Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space. Trans. Amer. Math. Soc., 361 11 (2009) pp. 5737-5752.
[DFM] B. Daniel, I. Fernández, P. Mira. The Gauss map of surfaces in PSL(2,R). Calc. var. partial differ. equ. 52 3 (2015), pp. 507-528.

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