81,408 research outputs found
On Asymptotic Weil-Petersson Geometry of Teichm\"{u}ller Space of Riemann Surfaces
In this paper, we study the asymptotic geometry of Teichmuller space of
Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of
Teichmuller space, in terms of the length of the shortest geodesic on the
surface. This will also imply that the sectional curvature is not pinched from
above or below by any negative constant.Comment: 38 pages, minor revisio
Asymptotic flatness of the Weil-Petersson metric on Teichmuller space
The sectional curvature of the Weil-Petersson metric on Teichmuller space is
known to be negative. We show that this Weil-Petersson sectional curvature is
not pinched from above by any negative constants, i.e., there is no negative
upper bound.Comment: 26 pages, to appear on Geometriae Dedicat
The Weil-Petersson Geometry On the Thick Part of the Moduli Space of Riemann Surfaces
On the thick part of the moduli space of Riemann surfaces, where there is a
positive lower bound of the systole of the surface, we show that all
Weil-Petersson Riemannian curvatures are bounded, independent of the genus of
the surface.Comment: 11 pages, a missing reference is adde
Do W_L and H form a p-wave bound state?
We examine the possibility of bound state formation in the W_L H --> W_L H
channel. The dynamical calculation using the N/D method indicates that when the
interactions among the Goldstone and Higgs bosons become sufficiently strong, a
p-wave state [I^G(J^P)=1^-(1^+)] may emerge.Comment: 3 pages, one separate uuencoded figure. Talk presented at Beyond the
Standard Model IV, 13-18 December 1994, Lake Tahoe
Asymptotics of the Gaussian Curvatures of the Canonical Metric on the Surface
We study the canonical metric on a compact Riemann surface of genus at least
two. While it is known that the canonical metric is of nonpositive curvature,
we show that its Gaussian curvatures are not bounded away from zero nor
negative infinity when the surface is close to the compactification divisor of
Riemann's moduli space.Comment: 11 pages. A previous post "The Canonical Metric on a Riemann Surface
and Its Induced Metric on Teichm\"{u}ller Space" has been rewritten to two
separate papers. This is the one focusing on the canonical metric on a
compact Riemann surfac
The Weil-Petersson curvature operator on the universal Teichm\"uller space
The universal Teichm\"uller space is an infinitely dimensional generalization
of the classical Teichm\"uller space of Riemann surfaces. It carries a natural
Hilbert structure, on which one can define a natural Riemannian metric, the
Weil-Petersson metric. In this paper we investigate the Weil-Petersson
Riemannian curvature operator of the universal Teichm\"uller space
with the Hilbert structure, and prove the following:
(i) is non-positive definite.
(ii) is a bounded operator.
(iii) is not compact; the set of the spectra of is
not discrete.
As an application, we show that neither the Quaternionic hyperbolic space nor
the Cayley plane can be totally geodesically immersed in the universal
Teichm\"uller space endowed with the Weil-Petersson metric.Comment: Math. Ann, to appea
Mean Curvature Flows in Almost Fuchsian Manifolds
An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold
that contains a closed incompressible minimal surface with principal curvatures
everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the
minimal surface is unique and embedded, hence one can parametrize these
three-manifolds by their minimal surfaces. We prove that any closed surface
which is a graph over any fixed surface of small principal curvatures can be
deformed into the minimal surface via the mean curvature flow. We also obtain
an upper bound for the hyperbolic volume of the convex core of M, as well as
estimates of the Hausdorff dimension of the limit set for .Comment: 22 pages. Sections are re-organized to clarify some arguments, a
subsection is added and some typos are fixed
Closed Minimal Surfaces in Cusped Hyperbolic Three-manifolds
Motivated by classical theorems on minimal surface theory in compact
hyperbolic three-manifolds, we investigate the questions of existence and
deformations for least area minimal surfaces in complete noncompact hyperbolic
three-manifold of finite volume. We prove any closed immersed incompressible
surface can be deformed to a closed immersed least area surface within its
homotopy class in any cusped hyperbolic three-manifold. Our techniques
highlight how special structures of these cusped hyperbolic three-manifolds
prevent any least area minimal surface going too deep into the cusped region.Comment: 23 pages, 2 figures: Final version, to appear in Geometriae Dedicat
Complex length of short curves and Minimal Fibration in hyperbolic -Manifolds fibering over the circle
We investigate the maximal solid tubes around short simple geodesics in
hyperbolic three-manifolds and how complex length of curves relate to closed,
incompressible, least area minimal surfaces. As applications, we prove, there
are some closed hyperbolic three-manifolds fibering over the circle which are
not foliated by closed incompressible minimal surfaces diffeomorphic to the
fiber. We also show, the existence of quasi-Fuchsian manifolds containing
arbitrarily many embedded closed incompressible minimal surfaces. Our strategy
is to prove main theorems under natural geometric conditions on the complex
length of closed curves on a fibered hyperbolic three-manifold, then we find
explicit examples where these conditions are satisfied via computer programs.Comment: final version, to appear at the Proceeding of the London Math. So
Counting Minimal Surfaces in Quasi-Fuchsian three-Manifolds
It is well known that every quasi-Fuchsian manifold admits at least one
closed incompressible minimal surface, and at most finitely many of them. In
this paper, for any prescribed integer , we construct a quasi-Fuchsian
manifold which contains at least such minimal surfaces. As a consequence,
there exists some simple close Jordan curve on such that there are
at least disk-type complete minimal surface in sharing
this Jordan curve as the asymptotic boundary.Comment: 22 pages, 9 figures, changes made following referee's many
corrections and suggestions. Accepted by the Transactions of the AM
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