My research concerns minimal and constant mean curvature surfaces in homogeneous three-manifolds. Construction and classification of such surfaces is a classical problem.

One particular case I am interested in are properly embedded annuli with constant mean curvature H > 1/2 in H^2 x R. Meeks conjectured that these surfaces are cylindrically bounded and singly-periodic with respect to translations along a geodesic in H^2 x R. These surfaces generalise unduloids from Euclidean space.

Hidden in this uniqueness conjecture is an existence problem: Given a geodesic in H^2 x R, is there a properly embedded CMC annulus with H > 1/2 that is periodic with respect to translations along that geodesic? For a vertical geodesic such surfaces arise as rotationally-invariant surfaces (vertical unduloids), for a horizontal geodesic it is sufficient to construct 1/4 of a horizontal unduloid via a conjugate Plateau construction (horizontal unduloids by Manzano and Torralbo). However, for a tilted geodesic no results were available.

In my Ph.D. thesis (see below) I studied this existence problem. A special case is to consider translationally-invariant annuli, which generalise cylinders. This yields surfaces in various ambient manifolds: tilted cylinders in H^2 x R, horizontal cylinders in PSL(2,R) and some cylinders in Sol. The general problem for tilted unduloids is reduced to a uniqueness problem in the Berger spheres.

As a post-doctoral researcher I found geodesic polygons in solve geometry Sol such that Schwarz reflection across its edges leads to triply periodic minimal surfaces. These are the first examples of such surfaces in Sol. It is joint work with Karsten Große-Brauckmann.

- "First examples of triply periodic embedded minimal surfaces in Sol" (with K. Große-Brauckmann), Preprint in arXiv
- "On the existence problem for tilted unduloids in H^2 x R", Revised preprint in arXiv, submission in progress
- "Cylinders as left invariant CMC surfaces in Sol and E(k,t)-spaces diffeomorphic to Euclidean three-space", Revised preprint in arXiv, Submission to Differential Geometry and its Applications in progress
- "Constant Mean Curvature Annuli in Homogeneous Manifolds", Publication of Ph.D. thesis on tuprints

- "Invited Talk at Universität Konstanz" July 14th 2016 in Constance
- "38. Süddeutsches Kolloquium über Differentialgeometrie" (in English: "38th Southern German Colloquium on Differential Geometry") on June 17th and June 18th 2016 in Mainz
- Participant of conference "Advances in Geometric Analysis" from July 20th to July 24th 2015 in Warwick
- Participant of Workshop "Flowers & Friends" on Geometric Analysis from March 2nd to March 5th 2015 in Frankfurt
- Research seminar DA-FF-MZ on Febraury 13th 2015 in Darmstadt
- "37. Süddeutsches Kolloquium über Differentialgeometrie" (in English: "37th Southern German Colloquium on Differential Geometry") on July 11th and July 12th 2014 in Ulm
- Participant of Workshop "Minimal Submanifolds and Related Topics" from August 26th to August 29th 2013 in Hannover
- "36. Süddeutsches Kolloquium über Differentialgeometrie" (in English: "36th Southern German Colloquium on Differential Geometry") on June 21st and 22nd 2013 in Constance
- Participant of Winter School "Geometric Evolution Equations and Related Topics" from October 8th to October 10th 2012 in Regensburg
- Participant of Workshop on "Geometric Analysis" from March 27th to March 30th 2012 in Frankfurt

Mail: vrzina