Realization of HCMU metrics in 3-dimensional space forms as Weingarten surfaces

TL;DR

This paper classifies HCMU surfaces in 3D space forms as Weingarten surfaces using a novel method, providing new criteria for when a surface is both HCMU and Weingarten.

Contribution

It introduces a different approach to classify HCMU surfaces as Weingarten surfaces in 3D space forms and establishes criteria for their intersection.

Findings

HCMU surfaces in 3D space forms are classified as Weingarten surfaces.

A new criterion for identifying surfaces that are both HCMU and Weingarten.

The method differs from previous classifications based on holomorphic functions.

Abstract

In \cite{WeiWu22-3} (J.Geom.Anal.32, 199(2022)), we classified HCMU surfaces in 3-dimensional Euclidean space as Weingarten surfaces by holomorphic functions. In this paper, using a totally different method from \cite{WeiWu22-3}, we will classify HCMU surfaces in 3-dimensional space forms as Weingarten surfaces. Moreover we will give a criteria of a Weingarten surface which is also an HCMU surface.