693 research outputs found
Classification of singularities in the complete conformally flat Yamabe flow
We show that an eternal solution to a complete, locally conformally flat
Yamabe flow, , with uniformly bounded
scalar curvature and positive Ricci curvature at , where the scalar
curvature assumes its maximum is a gradient steady soliton. As an application
of that, we study the blow up behavior of at the maximal time of
existence, . We assume that satisfies (i) the
injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time
and the scalar curvature bounded at each time-slice. We show that the
singularity the flow develops at time is always of type I.Comment: The paper has been withdrawn due to a crucial error in the argumen
The Yang-Mills flow near the boundary of Teichmueller space
We study the behavior of the Yang-Mills flow for unitary connections on
compact and non-compact oriented surfaces with varying metrics. The flow can be
used to define a one dimensional foliation on the space of SU(2)
representations of a once punctured surface. This foliation universalizes over
Teichm\"uller space and is equivariant with respect to the action of the
mapping class group. It is shown how to extend the foliation as a singular
foliation over the Strebel boundary of Teichm\"uller space, and continuity of
this extension is the main result of the paper.Comment: 35 pages. Late
The classification of locally conformally flat Yamabe solitons
We provide the classification of locally conformally flat gradient Yamabe
solitons with positive sectional curvature. We first show that locally
conformally flat gradient Yamabe solitons with positive sectional curvature
have to be rotationally symmetric and then give the classification and
asymptotic behavior of all radially symmetric gradient Yamabe solitons. We also
show that any eternal solutions to the Yamabe flow with positive Ricci
curvature and with the scalar curvature attaining an interior space-time
maximum must be a steady Yamabe soliton
Type II extinction profile of maximal solutions to the Ricci flow in
We consider the initial value problem ,
in , corresponding to the Ricci flow, namely
conformal evolution of the metric by Ricci curvature. It
is well known that the maximal (complete) solution vanishes identically
after time . Assuming that is
compactly supported we describe precisely the Type II vanishing behavior of
at time : we show the existence of an inner region with exponentially fast
vanishing profile, which is, up to proper scaling, a {\em soliton cigar
solution}, and the existence of an outer region of persistence of a logarithmic
cusp. This is the only Type II singularity which has been shown to exist, so
far, in the Ricci Flow in any dimension. It recovers rigorously formal
asymptotics derived by J.R. King \cite{K}
Eternal Solutions to the Ricci Flow on
We provide the classification of eternal (or ancient) solutions of the
two-dimensional Ricci flow, which is equivalent to the fast diffusion equation
on We show
that, under the necessary assumption that for every , the solution
defines a complete metric of bounded curvature and bounded width,
is a gradient soliton of the form , for some and some constants
and
Classification of Weil-Petersson Isometries
This paper contains two main results. The first is the existence of an
equivariant Weil-Petersson geodesic in Teichmueller space for any choice of
pseudo-Anosov mapping class. As a consequence one obtains a classification of
the elements of the mapping class group as Weil-Petersson isometries which is
parallel to the Thurston classification. The second result concerns the
asymptotic behavior of these geodesics. It is shown that geodesics that are
equivariant with respect to independent pseudo-Anosov's diverge. It follows
that subgroups of the mapping class group which contain independent
pseudo-Anosov's act in a reductive manner with respect to the Weil-Petersson
geometry. This implies an existence theorem for equivariant harmonic maps to
the metric completion.Comment: 31 pages; to appear in the American Journal of Mathematic
On the Brill-Noether Problem for Vector Bundles
On an arbitrary compact Riemann surface, necessary and sufficient conditions
are found for the existence of semistable vector bundles with slope between
zero and one and a prescribed number of linearly independent holomorphic
sections. Existence is achieved by minimizing the Yang-Mills-Higgs functional.Comment: LaTeX 2e (amsart
C^{1,\al} regularity of solutions to parabolic Monge-Amp\'ere equations
We study interior C^{1, \al} regularity of viscosity solutions of the
parabolic Monge-Amp\'ere equation u_t = b(x,t) \ddua, with exponent
and with coefficients which are bounded and measurable. We show that when
is less than the critical power then solutions become
instantly C^{1, \al} in the interior. Also, we prove the same result for any
power at those points where either the solution separates from the
initial data, or where the initial data is
Evolution of non-compact hypersurfaces by inverse mean curvature
We study the evolution of complete non-compact convex hypersurfaces in
by the inverse mean curvature flow. We establish the long
time existence of solutions and provide the characterization of the maximal
time of existence in terms of the tangent cone at infinity of the initial
hypersurface. Our proof is based on an a'priori pointwise estimate on the mean
curvature of the solution from below in terms of the aperture of a supporting
cone at infinity. The strict convexity of convex solutions is shown by means of
viscosity solutions. Our methods also give an alternative proof of the result
by Huisken and Ilmanen on compact start-shaped solutions, based on maximum
principle argument.Comment: 24 pages, 4 figure
On the extinction profile of solutions to fast-diffusion
We study the extinction behavior of solutions to the fast diffusion equation
on , in the range of exponents , . We show that if the initial data is trapped in
between two Barenblatt solutions vanishing at time , then the vanishing
behaviour of at is given by a Barenblatt solution. We also give an
example showing that for such a behavior the bound from above by a Barenblatt
solution (vanishing at ) is crucial: we construct a class of solutions
with initial data , near , which live longer
than and change behaviour at . The behavior of such solutions is
governed by up to , while for the solutions become
integrable and exhibit a different vanishing profile. For the Yamabe flow () the above means that these solutions develop a
singularity at time , when the Barenblatt solution disappears, and at
they immediately smoothen up and exhibit the vanishing profile of a sphere.
In the appendix we show how to remove the assumption on the bound on
from below by a Barenblatt
- β¦