21 research outputs found
On short time existence for the planar network flow
We prove the existence of the flow by curvature of regular planar networks
starting from an initial network which is non-regular. The proof relies on a
monotonicity formula for expanding solutions and a local regularity result for
the network flow in the spirit of B. White's local regularity theorem for mean
curvature flow. We also show a pseudolocality theorem for mean curvature flow
in any codimension, assuming only that the initial submanifold can be locally
written as a graph with sufficiently small Lipschitz constant.Comment: Final version, to appear in Journal of Differential Geometry. 51
page
Entropy and reduced distance for Ricci expanders
Perelman has discovered two integral quantities, the shrinker entropy \cW
and the (backward) reduced volume, that are monotone under the Ricci flow \pa
g_{ij}/\pa t=-2R_{ij} and constant on shrinking solitons. Tweaking some signs,
we find similar formulae corresponding to the expanding case. The {\it
expanding entropy} \ctW is monotone on any compact Ricci flow and constant
precisely on expanders; as in Perelman, it follows from a differential
inequality for a Harnack-like quantity for the conjugate heat equation, and
leads to functionals and . The {\it forward reduced volume}
is monotone in general and constant exactly on expanders.
A natural conjecture asserts that converges as to a
negative Einstein manifold in some weak sense (in particular ignoring
collapsing parts). If the limit is known a-priori to be smooth and compact,
this statement follows easily from any monotone quantity that is constant on
expanders; these include \Vol(g)/t^{n/2} (Hamilton) and
(Perelman), as well as our new quantities. In general, we show that if
\Vol(g) grows like (maximal volume growth) then \ctW,
and remain bounded (in their appropriate ways) for all time. We
attempt a sharp formulation of the conjecture
Rigidity of generic singularities of mean curvature flow
Shrinkers are special solutions of mean curvature flow (MCF) that evolve by
rescaling and model the singularities. While there are infinitely many in each
dimension, [CM1] showed that the only generic are round cylinders \SS^k\times
\RR^{n-k}. We prove here that round cylinders are rigid in a very strong
sense. Namely, any other shrinker that is sufficiently close to one of them on
a large, but compact, set must itself be a round cylinder.
To our knowledge, this is the first general rigidity theorem for
singularities of a nonlinear geometric flow. We expect that the techniques and
ideas developed here have applications to other flows.
Our results hold in all dimensions and do not require any a priori
smoothness.Comment: revised after acceptance for Publications IHE
The round sphere minimizes entropy among closed self-shrinkers
The entropy of a hypersurface is a geometric invariant that measures
complexity and is invariant under rigid motions and dilations. It is given by
the supremum over all Gaussian integrals with varying centers and scales. It is
monotone under mean curvature flow, thus giving a Lyapunov functional.
Therefore, the entropy of the initial hypersurface bounds the entropy at all
future singularities. We show here that not only does the round sphere have the
lowest entropy of any closed singularity, but there is a gap to the second
lowest
Entropy and reduced distance for Ricci expanders
Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij/∂t = − 2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals μ+ and v+. The forward reduced volume θ+ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t → ∞ to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known a-priori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/tn/2 (Hamilton) and -λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like tn/2(maximal volume growth) then W+, θ+ and -λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjectur
Symmetry of Traveling Wave Solutions to the Allen-Cahn Equation in \Er^2
In this paper, we prove even symmetry of monotone traveling wave solutions to
the balanced Allen-Cahn equation in the entire plane. Related results for the
unbalanced Allen-Cahn equation are also discussed