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arXiv · 2105.01887

Special Liouville metric with the Ricci condition

Katsuei Kenmotsu
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Abstract

Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dimensional Riemannian manifold with the special type of the Liouville metric satisfying the Ricci condition to the complex hyperbolic plane with parallel mean curvature vector.

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Katsuei Kenmotsu. 2021-05-05. Special Liouville metric with the Ricci condition. https://doi.org/10.1090/proc%2F15652

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