Related papers: Colding Minicozzi Entropy in Hyperbolic Space
Through direct thermodynamic calculations we have shown that different classical entropies of two-dimensional extreme black holes appear due to two different treatments, namely Hawking's treatment and Zaslavskii's treatment. Geometrical and…
We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In…
We present a coherent picture of the quantum mechanics of black holes. The picture does not require the introduction of any drastically new physical effect beyond what is already known; it arises mostly from synthesizing and…
We address the problem of defining the concept of entropy for anisotropic cosmological models. In particular, we analyze for the Bianchi I and V models the entropy which follows from postulating the validity of the laws of standard…
The recent developments in fluid/gravity correspondence give a new impulse to the study of fluid dynamics of supersymmetric theories. In that respect, the entropy current formalism requires some modifications in order to be adapted to…
In an earlier paper \cite{mazeng} the authors introduced the notion of curvature entropy, and proved the plane log-Minkowski inequality of curvature entropy under the symmetry assumption. In this paper we demonstrate the plane log-Minkowski…
Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed…
In \cite{CM5}, Colding and Minicozzi describe a type of compactness property possessed by sequences of embedded minimal surfaces in $\Real^3$ with finite genus and with boundaries going to $\infty$. They show that any such sequence either…
We study nonlinear hyperbolic conservation laws posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and defined from a prescribed flux field of n-forms depending on a parameter (the unknown variable), a class of…
We describe local similarities and global differences between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. We also describe how to solve global period problems for constant mean…
We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…
Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface…
We study the dynamics of entropy in a time dependent potential and explore how disorder influences this entropy flow. We show that disorder can trap entropy at the edge of the atomic cloud enabling a novel cooling method. We demonstrate the…
This is a review on entropy in various fields of mathematics and science. Its scope is to convey a unified vision of the classical as well as some newer entropy notions to a broad audience with an intermediate background in dynamical…
Entropy for two dimensional extremal black holes is computed explicitly in a finite-space formulation of the black hole thermodynamics and is shown to be zero {\it locally}. Our results are in conformity with the recent one by Hawking et al…
Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an…
We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of…
This paper investigates some properties of entropy solutions of hyperbolic conservation laws on a Riemannian manifold. First, we generalize the Total Variation Diminishing (TVD) property to manifolds, by deriving conditions on the flux of…
For inviscid fluid flow in any n-dimensional Riemannian manifold, new conserved vorticity integrals generalizing helicity, enstrophy, and entropy circulation are derived for lower-dimensional surfaces that move along fluid streamlines.…
We obtain explicit solutions of the mean curvature flow in some submanifolds of the Euclidean space. We give particularly an explicit solution of the flow of a hypersurface in the Lagrangian self-expander $L$ which is constructed in the…