27 research outputs found
Closed geodesics in Alexandrov spaces of curvature bounded from above
In this paper, we show a local energy convexity of maps into
spaces. This energy convexity allows us to extend Colding and
Minicozzi's width-sweepout construction to produce closed geodesics in any
closed Alexandrov space of curvature bounded from above, which also provides a
generalized version of the Birkhoff-Lyusternik theorem on the existence of
non-trivial closed geodesics in the Alexandrov setting.Comment: Final version, 22 pages, 2 figures, to appear in the Journal of
Geometric Analysi
Measurable versions of the LS category on laminations
We give two new versions of the LS category for the set-up of measurable
laminations defined by Berm\'udez. Both of these versions must be considered as
"tangential categories". The first one, simply called (LS) category, is the
direct analogue for measurable laminations of the tangential category of
(topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For
the measurable lamination that underlies any lamination, our measurable
tangential category is a lower bound of the tangential category. The second
version, called the measured category, depends on the choice of a transverse
invariant measure. We show that both of these "tangential categories" satisfy
appropriate versions of some well known properties of the classical category:
the homotopy invariance, a dimensional upper bound, a cohomological lower bound
(cup length), and an upper bound given by the critical points of a smooth
function.Comment: 22 page