arXiv subjects

Miroslav Vržina

Publications and source records attributed to Miroslav Vržina.

2 recordsLinked to original sources

arXiv2018

Cylinders as Left Invariant CMC Surfaces in $\operatorname{Sol}_3$ and $E(κ,τ)$-Spaces Diffeomorphic to $\mathbb{R}^3$

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature $H>H(X)$ in certain simply connected homogeneous three-manifolds $X$ diffeomorphic to $\mathbb{R}^3$, which always admit a Lie group structure. Here, $H(X)$ denotes the critical value for which constant mean curvature spheres in $X$ exist. Our cylinders are generated by a simple closed curve under a one-parameter group of isometries, induced by left translations along certain geodesics. In the spaces $\operatorname{Sol}_3$ and $\widetilde{\operatorname{PSL}}_2(\mathbb{R})$ we establish existence of new properly embedded constant mean curvature annuli. We include computed examples of cylinders in $\operatorname{Sol}_3$ generated by non-embedded simple closed curves.

math.DG↗

arXiv2017

On the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$

We study the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$. These are singly periodic annuli with constant mean curvature $H>1/2$ in $\mathbb{H}^2\times\mathbb{R}$, and the periodicity of these surfaces is with respect to a discrete group of translations along a geodesic that is neither vertical nor horizontal in the Riemannian product $\mathbb{H}^2\times\mathbb{R}$. Via the Daniel correspondence we are able to reduce this existence problem to a uniqueness problem in the Berger spheres: if a pair of linked horizontal geodesics bounds exactly two embedded minimal annuli (for a fixed orientation of the boundary curves) then tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$ exist.

math.DG↗