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A classification of constant Gaussian curvature surfaces in the three-dimensional hyperbolic space
Published 12 Apr 2024 in math.DG | (2404.08235v1)
Abstract: Weakly complete constant Gaussian curvature $-1<K\<0$ surfaces will be classified in terms of holomorphic quadratic differentials. For this purpose, we will first establish a loop group method for constant Gaussian curvature surfaces in $\mathbb H^3$ with $K>-1$ but $K \neq 0$ by using harmonicities of Lagrangian and Legendrian Gauss maps. Then we will show that a spectral parameter deformation of the Lagrangian harmonic Gauss map gives a harmonic map into $\mathbb H2$ for $-1< K<0$ or $\mathbb S2$ for $K>0$, respectively.
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