1,466 research outputs found
A Zoll counterexample to a geodesic length conjecture
We construct a counterexample to a conjectured inequality L<2D, relating the
diameter D and the least length L of a nontrivial closed geodesic, for a
Riemannian metric on the 2-sphere. The construction relies on Guillemin's
theorem concerning the existence of Zoll surfaces integrating an arbitrary
infinitesimal odd deformation of the round metric. Thus the round metric is not
optimal for the ratio L/D.Comment: 10 pages; to appear in Geometric and Functional Analysi
Some Lipschitz maps between hyperbolic surfaces with applications to Teichmüller theory
International audienceIn the Teichmüller space of a hyperbolic surface of finite type, we construct geodesic lines for Thurston's asymmetric metric having the property that when they are traversed in the reverse direction, they are also geodesic lines (up to reparametrization). The lines we construct are special stretch lines in the sense of Thurston. They are directed by complete geodesic laminations that are not chain-recurrent, and they have a nice description in terms of Fenchel-Nielsen coordinates. At the basis of the construction are certain maps with controlled Lipschitz constants between right-angled hyperbolic hexagons having three non-consecutive edges of the same size. Using these maps, we obtain Lipschitz-minimizing maps between hyperbolic particular pairs of pants and, more generally, between some hyperbolic sufaces of finite type with arbitrary genus and arbitrary number of boundary components. The Lipschitz-minimizing maps that we contruct are distinct from Thurston's stretch maps
Constrained deformations of positive scalar curvature metrics
We present a series of results concerning the interplay between the scalar
curvature of a manifold and the mean curvature of its boundary. In particular,
we give a complete topological characterization of those compact 3-manifolds
that support Riemannian metrics of positive scalar curvature and mean-convex
boundary and, in any such case, we prove that the associated moduli space of
metrics is path-connected. The methods we employ are flexible enough to allow
the construction of continuous paths of positive scalar curvature metrics with
minimal boundary, and to derive similar conclusions in that context as well.
Our work relies on a combination of earlier fundamental contributions by
Gromov-Lawson and Schoen-Yau, on the smoothing procedure designed by Miao, and
on the interplay of Perelman's Ricci flow with surgery and conformal
deformation techniques introduced by Cod\'a Marques in dealing with the closed
case.Comment: 80 pages; final pre-print version, accepted for publication in JD
A Minimal Length from the Cutoff Modes in Asymptotically Safe Quantum Gravity
Within asymptotically safe Quantum Einstein Gravity (QEG), the quantum
4-sphere is discussed as a specific example of a fractal spacetime manifold.
The relation between the infrared cutoff built into the effective average
action and the corresponding coarse graining scale is investigated. Analyzing
the properties of the pertinent cutoff modes, the possibility that QEG
generates a minimal length scale dynamically is explored. While there exists no
minimal proper length, the QEG sphere appears to be "fuzzy" in the sense that
there is a minimal angular separation below which two points cannot be resolved
by the cutoff modes.Comment: 26 pages, 1 figur
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