48 research outputs found
Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group
We study symmetric minimal surfaces in the three-dimensional Heisenberg group
using the generalized Weierstrass type representation, the
so-called loop group method. In particular, we will discuss how to construct
minimal surfaces in with non-trivial topology. Moreover, we
will classify equivariant minimal surfaces given by one-parameter subgroups of
the isometry group of .Comment: 49 page
The Bour's Theorem for invariant surfaces in three-manifolds
In this paper, we apply techniques of the equivariant geometry to give a
positive answer to the conjecture that a generalized Bour's Theorem holds for
surfaces that are invariant under the action of a one-parameter group of
isometries of a three-dimensional Riemannian manifold.Comment: 17 page
Minding isometries of ruled surfaces in Lorentz-Minkowski space
In this paper we study isometries of ruled surfaces in the Lorentz-Minkowski space that preserve rulings. A special attention is given to the classes of surfaces having no Euclidean counterparts. We also construct some examples of isometric ruled surfaces with certain properties and rulings preserved
Generalized Helical Hypersurfaces Having Time-like Axis in Minkowski Spacetime
In this paper, the generalized helical hypersurfaces x=x(u,v,w) with a time-like axis in Minkowski spacetime E14 are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of x are calculated. Moreover, the curvatures of the generalized helical hypersurface x are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of x are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in E14
Blackfolds, Plane Waves and Minimal Surfaces
Minimal surfaces in Euclidean space provide examples of possible non-compact
horizon geometries and topologies in asymptotically flat space-time. On the
other hand, the existence of limiting surfaces in the space-time provides a
simple mechanism for making these configurations compact. Limiting surfaces
appear naturally in a given space-time by making minimal surfaces rotate but
they are also inherent to plane wave or de Sitter space-times in which case
minimal surfaces can be static and compact. We use the blackfold approach in
order to scan for possible black hole horizon geometries and topologies in
asymptotically flat, plane wave and de Sitter space-times. In the process we
uncover several new configurations, such as black helicoids and catenoids, some
of which have an asymptotically flat counterpart. In particular, we find that
the ultraspinning regime of singly-spinning Myers-Perry black holes, described
in terms of the simplest minimal surface (the plane), can be obtained as a
limit of a black helicoid, suggesting that these two families of black holes
are connected. We also show that minimal surfaces embedded in spheres rather
than Euclidean space can be used to construct static compact horizons in
asymptotically de Sitter space-times.Comment: v2: 67pp, 7figures, typos fixed, matches published versio