Related papers: The Log Entropy Functional Along the Ricci Flow

In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak…

We consider a new functional inequality controlling the rate of relative entropy decay for random walks, the interchange process and more general block-type dynamics for permutations. The inequality lies between the classical logarithmic…

In this paper, we introduce a new notion of curvature on the edges of a graph that is defined in terms of effective resistances. We call this the Ricci--Foster curvature. We study the Ricci flow resulting from this curvature. We prove the…

In the article, we generalize some recent results of Colding and Minicozzi on generic singularities of mean curvature flow to curved ambient spaces. To do so, we make use of a weighted monotonicity formula to derive an "almost monotonicity"…

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the…

The present paper studies continuity of generalized entropy functions and relative entropies defined using the notion of a deformed logarithmic function. In particular, two distinct definitions of relative entropy are discussed. As an…

We prove an optimal relative integral convergence rate for two expanding gradient Ricci solitons coming out of the same cone. As a consequence, we obtain a unique continuation result at infinity and we prove that a relative entropy for two…

In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a…

We study the Ricci flow on Riemannian groupoids. We assume that these groupoids are closed and that the space of orbits is compact and connected. We prove the short time existence and uniqueness of the Ricci flow on these groupoids. We also…

Entropy rate is a real valued functional on the space of discrete random sources which lacks a closed formula even for subclasses of sources which have intuitive parameterizations. A good way to overcome this problem is to examine its…

The famous Uniformization Theorem states that on closed Riemannian surfaces there always exists a metric of constant curvature for the Levi-Cevita connection. In this article we prove that an analogue of the uniformization theorem also…

We study monotonic quantities in the context of combined geometric flows. In particular, focusing on Ricci solitons as the ambient space, we consider solutions of the heat type equation integrated over embedded submanifolds evolving by mean…

In this paper, we study monotonicity of eigenvalues of Laplacian-type operator $-\Delta+cR$, where $c$ is a constant, along the Ricci-Bourguignon flow. For $c\neq0$, We derive monotonicity of the lowest eigenvalue of Laplacian-type operator…

In this paper we present some results on a family of geometric flows introduced by Bourguignon that generalize the Ricci flow. For suitable values of the scalar parameter involved in these flows, we prove short time existence and provide…

In this note we explain how a flow in the space of Riemmanian metrics (including Ricci's \cite{mt}) induces one in the space of pseudoconnections.

We study some asymptotic behavior of the first nonzero eigenvalue of the Lalacian along the normalized Ricci flow and give a direct short proof for an asymptotic upper limit estimate.

In this paper, we consider functionals related to mean curvature flow in an ambient space which evolves by an extended Ricci flow from the perspective introduced by Lott when studying a mean curvature flow in a Ricci flow background. One of…

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We…

Entropy functionals are computed for non-stationary distributions of particles of Lorentz gas and hard disks. The distributions consisting of beams of particles are found to have the largest amount of entropy and entropy increase. The…

We revisit the existence of monotonic quantities along renormalization group flows using only the Null Energy Condition and the Ryu-Takayanagi formula for the entanglement entropy of field theories with anti-de Sitter gravity duals. In…