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

Naoya Ando, Masaaki Umehara

TL;DR

This paper analyzes time-like constant mean curvature surfaces in Lorentzian 3-space forms by linking curvature line flows to the Hopf differential using a para-holomorphic (paracomplex) framework. It defines the Hopf differential $Q_f$ and its finite-type, non-degenerate structure, showing that the local behavior of curvature line flows near an isolated umbilic is governed by the zero order $m$ of $Q_f$ modulo 4, with indices 0 or $\pm1$ depending on $m$. It also proves that isolated umbilics are accumulation points of quasi-umbilics and provides explicit ZMC constructions to illustrate the index phenomena, while contrasting the time-like case with the space-like case via Appendix results. The work highlights subtle geometric structures unique to time-like CMC surfaces and offers tools for analyzing curvature line configurations in Lorentzian geometry.

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.

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

TL;DR

This paper analyzes time-like constant mean curvature surfaces in Lorentzian 3-space forms by linking curvature line flows to the Hopf differential using a para-holomorphic (paracomplex) framework. It defines the Hopf differential and its finite-type, non-degenerate structure, showing that the local behavior of curvature line flows near an isolated umbilic is governed by the zero order of modulo 4, with indices 0 or depending on . It also proves that isolated umbilics are accumulation points of quasi-umbilics and provides explicit ZMC constructions to illustrate the index phenomena, while contrasting the time-like case with the space-like case via Appendix results. The work highlights subtle geometric structures unique to time-like CMC surfaces and offers tools for analyzing curvature line configurations in Lorentzian geometry.

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.

Paper Structure

This paper contains 5 sections, 12 theorems, 71 equations.

Table of Contents

  1. Preliminaries
  2. The Hopf differentials of time-like CMC surfaces
  3. Curvature line flows around umbilics
  4. Space-like CMC surfaces
  5. Properties of smooth para-holomorphic functions

Key Result

Proposition 1.2

Let $p\in U$ be an isolated umbilic of a time-like surface $f:U\to M^3_1$, and let ${\mathcal{F}}$ be a $C^r$ curvature-line flow on $U\setminus \{p\}$, where $r\ge 1$. Then there exists a unique $C^r$ curvature-line flow ${\mathcal{F}}^\perp$ on $U\setminus \{p\}$ such that

Theorems & Definitions (39)

  • Definition 1
  • Remark 1.1
  • Proposition 1.2
  • proof
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 29 more