Biconservative surfaces in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$
Dorel Fetcu
TL;DR
This work addresses the problem of explicitly describing non-constant mean curvature biconservative surfaces in the product spaces $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ under the assumption that the gradient of the mean curvature, $\nabla f$, is a principal direction. Building on the tangent-part vanishing condition for the biharmonic (bienergy) framework, the authors derive a reduced structure governed by a pair of functions $a(u)$ and $\theta(u)$, and obtain explicit local parametrizations. In $\mathbb{S}^2\times\mathbb{R}$ they provide a single explicit local model in $\mathbb{R}^4$ with $\Phi(u,v)$ given by $\displaystyle \left(\frac{\cos v}{\sqrt{a^2+1}}C_1+\frac{\sin v}{\sqrt{a^2+1}}C_2+\frac{a}{\sqrt{a^2+1}}(C_1\times C_2),\int_{u_0}^u\cos\theta(t)dt\right)$ and the ODEs $a'(u)=-(a^2+1)\sin\theta$ along with a second constraint on $\theta$. In $\mathbb{H}^2\times\mathbb{R}$ they classify three local families according to $a^2=1$, $a^2>1$, and $a^2<1$, each yielding explicit $\Phi(u,v)$ formulae in $\mathbb{L}^3\times\mathbb{R}$ and, when appropriate, in Lorentzian settings, with the same governing differential relations for $a$ and $\theta$. These results extend the catalog of explicit non-CMC biconservative surfaces beyond space forms, clarifying the role of the principal-direction gradient in product geometries and enabling concrete constructions in two Thurston product geometries.
Abstract
We find the explicit local equations of biconservative surfaces with non-constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, when the gradient of the mean curvature function is a principal direction.