Isoparametric submanifolds in two-dimensional complex space forms

TL;DR

This paper classifies and characterizes isoparametric submanifolds in two-dimensional complex space forms, revealing their relation to polar actions and introducing a broader class of Terng-isoparametric submanifolds.

Contribution

It provides a classification of Terng-isoparametric submanifolds and clarifies their relationship with polar actions in complex hyperbolic planes.

Findings

Isoparametric submanifolds are open parts of principal orbits of polar actions.

Existence of non-isoparametric submanifolds that are Terng-isoparametric.

Complete classification of Terng-isoparametric submanifolds in two-dimensional complex space forms.

Abstract

We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos', is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the definition of Terng's. Finally, we classify Terng-isoparametric submanifolds of two-dimensional complex space forms.