81 research outputs found
A geometric study of marginally trapped surfaces in space forms and Robertson-Walker spacetimes -- an overview
A marginally trapped surface in a spacetime is a Riemannian surface whose
mean curvature vector is lightlike at every point. In this paper we give an
up-to-date overview of the differential geometric study of these surfaces in
Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give
the general local descriptions proven by Anciaux and his coworkers as well as
the known classifications of marginally trapped surfaces satisfying one of the
following additional geometric conditions: having positive relative nullity,
having parallel mean curvature vector field, having finite type Gauss map,
being invariant under a one-parameter group of ambient isometries, being
isotropic, being pseudo-umbilical. Finally, we provide examples of constant
Gaussian curvature marginally trapped surfaces and state some open questions.Comment: 21 page
Sequences of harmonic maps in the 3-sphere
We define two transforms between non-conformal harmonic maps from a surface
into the 3-sphere. With these transforms one can construct, from one such
harmonic map, a sequence of harmonic maps. We show that there is a
correspondence between non-conformal harmonic maps into the 3-sphere,
-surfaces in Euclidean 3-space and almost complex surfaces in the nearly
K\"ahler manifold . As a consequence we can construct sequences
of -surfaces and almost complex surfaces.Comment: 14 pages. Second version. The article has been extended and is
thoroughly revise
Pseudo-parallel Lagrangian submanifolds are semi-parallel
We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in
[Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential
Geom. Appl.] stating that every Lagrangian pseudo-parallel submanifold of a
complex space form of dimension at least 3 is semi-parallel
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