Hopf differentials and curvature line flows on time-like CMC surfaces

TL;DR

This paper explores the connection between Hopf differentials and curvature line flows on time-like CMC surfaces in Lorentzian space, revealing how the differential's properties influence flow indices at umbilic points.

Contribution

It establishes a precise relationship between Hopf differentials and curvature line flow indices, especially when the differential is non-degenerate.

Findings

Flow index at umbilic points depends on the order of Hopf differential modulo four.

Non-degenerate Hopf differential leads to specific curvature flow behaviors.

Provides new insights into geometric structures of time-like CMC surfaces.

Abstract

We investigate the relationship between the Hopf differentials and the curvature line flows on time-like constant mean curvature (CMC) surfaces in Lorentzian 3-space forms. In particular, when the Hopf differential is non-degenerate, the index of a curvature line flow at an umbilic point depends precisely on the remainder of its order modulo four.