Double CMC surfaces in Euclidean and hyperbolic spaces
Volume 129 / 2022
Annales Polonici Mathematici 129 (2022), 139-145
MSC: Primary 53C42; Secondary 53A35.
DOI: 10.4064/ap211015-12-7
Published online: 1 August 2022
Abstract
We establish a rigidity theorem for $2$-manifolds immersed in Euclidean and hyperbolic spaces, which have constant mean curvatures with respect to both induced metrics. Using the relation between the two mean curvatures, we prove that the two immersions are both totally umbilical. Moreover, we classify totally umbilical surfaces in $\mathbb R^3$ and $\mathbb H^3$ with respect to the value of the mean curvatures.
