185 research outputs found
A new notion of angle between three points in a metric space
We give a new notion of angle in general metric spaces; more precisely, given
a triple a points in a metric space , we introduce the notion of
angle cone as being an interval
, where the quantities
are defined in terms of the distance functions from and
via a duality construction of differentials and gradients holding for
locally Lipschitz functions on a general metric space. Our definition in the
Euclidean plane gives the standard angle between three points and in a
Riemannian manifold coincides with the usual angle between the geodesics, if
is not in the cut locus of or . We show that in general the angle
cone is not single valued (even in case the metric space is a smooth Riemannian
manifold, if is in the cut locus of or ), but if we endow the metric
space with a positive Borel measure obtaining the metric measure space
then under quite general assumptions (which include many fundamental
examples as Riemannian manifolds, finite dimensional Alexandrov spaces with
curvature bounded from below, Gromov-Hausdorff limits of Riemannian manifolds
with Ricci curvature bounded from below, and normed spaces with strictly convex
norm), fixed , the angle cone at is single valued for -a.e.
. We prove some basic properties of the angle cone (such as the
invariance under homotheties of the space) and we analyze in detail the case
is a measured-Gromov-Hausdorff limit of a sequence of Riemannian
manifolds with Ricci curvature bounded from below, showing the consistency of
our definition with a recent construction of Honda.Comment: 19 page
On the universal cover and the fundamental group of an -space
The main goal of the paper is to prove the existence of the universal cover
for -spaces. This generalizes earlier work of C. Sormani and the
second named author on the existence of universal covers for Ricci limit
spaces. As a result, we also obtain several structure results on the (revised)
fundamental group of such spaces. These are the first topological results for
-spaces without extra structural-topological assumptions (such as
semi-local simple connectedness).Comment: Final version to appear in Journal f\"ur die Reine und Angewandte
Mathemati
Willmore Spheres in Compact Riemannian Manifolds
The paper is devoted to the variational analysis of the Willmore, and other
L^2 curvature functionals, among immersions of 2-dimensional surfaces into a
compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is
twofold, on one hand, we give the right setting for doing the calculus of
variations (including min max methods) of such functionals for immersions into
manifolds and, on the other hand, we prove existence results for possibly
branched Willmore spheres under various constraints (prescribed homotopy class,
prescribed area) or under curvature assumptions for M^m. To this aim, using the
integrability by compensation, we develop first the regularity theory for the
critical points of such functionals. We then prove a rigidity theorem
concerning the relation between CMC and Willmore spheres. Then we prove that,
for every non null 2-homotopy class, there exists a representative given by a
Lipschitz map from the 2-sphere into M^m realizing a connected family of
conformal smooth (possibly branched) area constrained Willmore spheres (as
explained in the introduction, this comes as a natural extension of the minimal
immersed spheres in homotopy class constructed by Sacks and Uhlembeck in
\cite{SaU}, in situations when they do not exist). Moreover, for every A>0 we
minimize the Willmore functional among connected families of weak, possibly
branched, immersions of the 2-sphere having prescribed total area equal to A
and we prove full regularity for the minimizer. Finally, under a mild curvature
condition on (M^m,h), we minimize the sum of the area with the square of the
L^2 norm of the second fundamental form, among weak possibly branched
immersions of the two sphere and we prove the regularity of the minimizer.Comment: 58 page
Immersed Spheres of Finite Total Curvature into Manifolds
We prove that a sequence of possibly branched, weak immersions of the
two-sphere into an arbitrary compact riemannian manifold with
uniformly bounded area and uniformly bounded norm of the second
fundamental form either collapse to a point or weakly converges as current,
modulo extraction of a subsequence, to a Lipschitz mapping of and whose
image is made of a connected union of finitely many, possibly branched, weak
immersions of with finite total curvature. We prove moreover that if the
sequence belongs to a class of the limiting lipschitz
mapping of realizes this class as well.Comment: 33 pages. Original preprint (2011). This is the final version to
appear in Adv. Calc. Va
Polya-Szego inequality and Dirichlet -spectral gap for non-smooth spaces with Ricci curvature bounded below
We study decreasing rearrangements of functions defined on (possibly
non-smooth) metric measure spaces with Ricci curvature bounded below by
and dimension bounded above by in a synthetic sense, the so
called spaces. We first establish a Polya-Szego type inequality
stating that the -Sobolev norm decreases under such a rearrangement
and apply the result to show sharp spectral gap for the -Laplace operator
with Dirichlet boundary conditions (on open subsets), for every . This extends to the non-smooth setting a classical result of
B\'erard-Meyer and Matei; remarkable examples of spaces fitting out framework
and for which the results seem new include: measured-Gromov Hausdorff limits of
Riemannian manifolds with Ricci, finite dimensional Alexandrov spaces
with curvature, Finsler manifolds with Ricci. In the second
part of the paper we prove new rigidity and almost rigidity results attached to
the aforementioned inequalities, in the framework of spaces, which
seem original even for smooth Riemannian manifolds with Ricci.Comment: 33 pages. Final version published in Journal de Math\'ematiques Pures
et Appliqu\'ee
On the volume measure of non-smooth spaces with Ricci curvature bounded below
We prove that, given an -space , then it is possible
to -essentially cover by measurable subsets
with the following property: for each there exists such that is absolutely continuous
with respect to the -dimensional Hausdorff measure. We also show that a
Lipschitz differentiability space which is bi-Lipschitz embeddable into a
euclidean space is rectifiable as a metric measure space, and we conclude with
an application to Alexandrov spaces.Comment: Final version to appear in the Annali della Scuola Normale Superiore
Classe di Scienz
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