CMC biconservative surfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$

TL;DR

This paper classifies certain biconservative surfaces with parallel mean curvature in product spaces and provides explicit equations for non-trivial cases, also discussing compactness in Hadamard manifolds.

Contribution

It offers a classification of non-minimal biconservative surfaces with parallel mean curvature in specific product spaces and derives explicit local equations for complex cases.

Findings

Explicit local equations for non-trivial biconservative surfaces.

Classification of non-minimal biconservative surfaces in product spaces.

Compactness results for biconservative surfaces in Hadamard manifolds.

Abstract

We classify non-minimal biconservative surfaces with parallel mean curvature vector field in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$. When these surfaces do not lie in $\mathbb{S}^n$ or $\mathbb{H}^n$ and they are not vertical cylinders, we find their explicit (local) equation. We also prove a result on the compactness of biconservative surfaces with constant mean curvature in Hadamard manifolds.