Area of research

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.


Publications (as of Ocotber 1st 2017)



Miroslav Vrzina

Mail: vrzina

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