Special Liouville metric with the Ricci condition

TL;DR

This paper explores special Liouville metrics satisfying Ricci-like conditions on surfaces, characterizing their explicit forms via elliptic functions and constructing isometric immersions into complex hyperbolic space.

Contribution

It identifies necessary conditions for metrics of parallel mean curvature surfaces and explicitly constructs examples satisfying these conditions.

Findings

Liouville metrics satisfying Ricci conditions are explicitly characterized by elliptic functions.

Isometric immersions into complex hyperbolic space are constructed for these metrics.

The work links classical surface theory conditions with complex space form geometry.

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.