Related papers: A heat flow for special metrics

Deforming a two dimensional conformal field theory on one side of a trivial defect line gives rise to a defect separating the original theory from its deformation. The Casimir force between these defects and other defect lines or boundaries…

We study deformations of Riemannian metrics on a given manifold equipped with a codimension-one foliation subject to quantities expressed in terms of its second fundamental form. We prove the local existence and uniqueness theorem and…

The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under…

We confirm a conjecture of Hamilton: On compact manifolds the normalized Ricci flow evolves metrics with positive curvature operators to limit metrics with constant curvature.

We present two geometric applications of heat flow methods on the discrete hypercube $\{-1,1\}^n$. First, we prove that if $X$ is a finite-dimensional normed space, then the bi-Lipschitz distortion required to embed $\{-1,1\}^n$ equipped…

We consider a transversally conformal foliation $\mathcal{F}$ of a closed manifold $M$ endowed with a smooth Riemannian metric whose restriction to each leaf is negatively curved. We prove that it satisfies the following dichotomy. Either…

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the…

This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the (4 + 1)-dimensional Minkowski…

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is…

Given a generic 2-plane field on a 5-dimensional manifold we consider its (3,2)-signature conformal metric [g] as defined in math.DG/0406400. Every conformal class [g] obtained in this way has very special conformal holonomy: it must be…

We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex…

We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also…

Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of…

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we…

In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution…

Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that…

Motivated by a paper of Bolsinov and Taimanov DG/9911193 we consider non-holonomic situation and exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. Moreover the Riemannian examples…

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry…

Chow and Hamilton introduced the cross curvature flow on closed 3-manifolds with negative or positive sectional curvature. In this paper, we study the negative cross curvature flow in the case of locally homogenous metrics on 3-manifolds.…

We consider regular open curves in R^n with fixed boundary points and moving according to the L^{2}-gradient flow for a generalisation of the Helfrich functional. Natural boundary conditions are imposed along the evolution. More precisely,…