Related papers: The round sphere minimizes entropy among closed se…
We propose an operational definition of the entropy of cosmological perturbations based on a truncation of the hierarchy of Green functions. The value of the entropy is unambiguous despite gauge invariance and the renormalization procedure.…
Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic K\"ahler manifold $(Y,g)$ and a compact complex {hyperbolic manifold} $(X,g_0)$. In this paper we give a lower bound of the diastatic entropy of $(Y,g)$ in terms…
In any static spacetime the quasilocal Tolman mass contained within a volume can be reduced to a Gauss-like surface integral involving the flux of a suitably defined generalized surface gravity. By introducing some basic thermodynamics, and…
In this paper we exhibit deformations of the hemisphere $S^{n+1}_+$, $n\geq 2$, for which the ambient Ricci curvature lower bound $\text{Ric}\geq n $ and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the…
The quasi-local notion of an isolated horizon is employed to study the entropy of black holes without any particular symmetry in loop quantum gravity. The idea of characterizing the shape of a horizon by a sequence of local areas is…
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain…
Entropy of matter in a very strong gravity depends on cross-sectional area of the container of the system -- is being further bolstered by calculating entropy of a monoatomic gas kept under uniform strong gravity at Newtonian scale. This…
We argue that every finite piecewise smooth spacelike two-surface of spacetime possesses an entropy which is, in natural units, one quarter of its area. However, the thermodynamical properties of a two-surface become apparent only to the…
Invariance entropy is a measure for the smallest data rate in a noiseless digital channel above which a controller that only receives state information through this channel is able to render a given subset of the state space invariant. In…
We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out…
When the Ricci curvature of a Riemannian manifold is not lower bounded by a constant, but lower bounded by a continuous function, we give a new characterization of this lower bound through the convexity of relative entropy on the…
We consider the situation when a globally defined four-dimensional field system is separated on two entangled sub-systems by a dynamical (random) two-dimensional surface. The reduced density matrix averaged over ensemble of random surfaces…
For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the…
We are concerned with hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold.…
We study the entanglement entropy in confining theories with gravity duals using the holographic prescription of Ryu and Takayanagi. The entanglement entropy between a region and its complement is proportional to the minimal area of a bulk…
The coefficient of the log term in the entanglement entropy associated with hyperspherical surfaces in flat space-time is shown to equal the conformal anomaly by conformally transforming Euclideanised space--time to a sphere and using…
We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes…
We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces…
We define a Bowen-Series like map for every geometric presentation of a co-compact surface group and we prove that the volume entropy of the presentation is the topological entropy of this particular (circle) map. Finally we find the…
We consider curve shortening flow of arbitrary codimension in an Euclidean background. We show that, close to a singularity, the flow is asymptotically planar, paralleling Altschuler's work in the case of space curves, and analyse the…