41 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
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
Strong deformation retraction of the space of Zoll Finsler projective planes
We show that the infinite-dimensional space of Zoll Finsler metrics on the
projective plane strongly deformation retracts to the canonical round metric.
In particular, this space of Zoll Finsler metrics is connected. Moreover, the
strong deformation retraction arises from a deformation of the geodesic flow of
every Zoll Finsler projective plane to the geodesic flow of the round metric
through a family of smooth free circle actions induced by the curvature flow of
the canonical round projective plane. This construction provides a description
of the geodesics of the Zoll Finsler metrics along the retraction