80 research outputs found
Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature
We establish a min-max estimate on the volume width of a closed Riemannian
manifold with nonnegative Ricci curvature. More precisely, we show that every
closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse
function whose level set volume is bounded in terms of the volume of the
manifold. As a consequence of this sweep-out estimate, there exists an
embedded, closed (possibly singular) minimal hypersurface whose volume is
bounded in terms of the volume of the manifold.Comment: 17 pages, 1 figur
Growth of quotients of groups acting by isometries on Gromov hyperbolic spaces
We show that every non-elementary group acting properly and cocompactly
by isometries on a proper geodesic Gromov hyperbolic space is growth tight.
In other words, the exponential growth rate of for the geometric
(pseudo)-distance induced by is greater than the exponential growth rate of
any of its quotients by an infinite normal subgroup. This result generalizes
from a unified framework previous works of Arzhantseva-Lysenok and Sambusetti,
and provides an answer to a question of the latter
Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic
Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat
1-unit-side equilateral triangles glued along their boundaries is a local
extremum for the length of the shortest closed geodesic among the Riemannian
spheres with conical singularities of fixed area. We give an alternative proof
of this theorem, which does not make use of the uniformization theorem, and
extend the result to Finsler metrics
Systolic volume and minimal entropy of aspherical manifolds
We establish isosystolic inequalities for a class of manifolds which includes
the aspherical manifolds. In particular, we relate the systolic volume of
aspherical manifolds first to their minimal entropy, then to the algebraic
entropy of their fundamental groups.Comment: 24 pages. J. Diff. Geom, to appea
An optimal systolic inequality for CAT(0) metrics in genus two
We prove an optimal systolic inequality for CAT(0) metrics on a genus~2
surface. We use a Voronoi cell technique, introduced by C.~Bavard in the
hyperbolic context. The equality is saturated by a flat singular metric in the
conformal class defined by the smooth completion of the curve y^2=x^5-x. Thus,
among all CAT(0) metrics, the one with the best systolic ratio is composed of
six flat regular octagons centered at the Weierstrass points of the Bolza
surface.Comment: 14 pages, 1 figure, to appear in Pacific J. Mat
Mixed sectional-Ricci curvature obstructions on tori
We establish new obstruction results to the existence of Riemannian metrics
on tori satisfying mixed bounds on both their sectional and Ricci curvatures.
More precisely, from Lohkamp's theorem, every torus of dimension at least three
admits Riemannian metrics with negative Ricci curvature. We show that the
sectional curvature of these metrics cannot be bounded from above by an
arbitrarily small positive constant. In particular, if the Ricci curvature of a
Riemannian torus is negative, bounded away from zero, then there exist some
planar directions in this torus where the sectional curvature is positive,
bounded away from zero. All constants are explicit and depend only on the
dimension of the torus
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