Surfaces with parallel mean curvature in Sasakian space forms

TL;DR

This paper investigates the geometry of surfaces with parallel mean curvature in Sasakian space forms, introducing new tools like holomorphic differentials to classify such surfaces and proving a codimension reduction theorem.

Contribution

It provides new classification results for surfaces with parallel mean curvature in Sasakian space forms and introduces holomorphic quadratic differentials for anti-invariant surfaces.

Findings

Classification theorems for surfaces with parallel mean curvature

Introduction of holomorphic quadratic differentials

Codimension reduction theorem

Abstract

We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension $2n+1$). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surfaces and use them to obtain classification theorems.