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 $\mu_+$ and $\nu_+$. The {\it forward reduced volume} $\theta_+$ is monotone in general and constant exactly on expanders. A natural conjecture asserts that $g(t)/t$ converges as $t\to\infty$ 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 $\bar\lambda$ (Perelman), as well as our new quantities. In general, we show that if \Vol(g) grows like $t^{n/2}$ (maximal volume growth) then \ctW, $\theta_+$ and $\bar\lambda$ 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