arXiv research

Optimal geometric estimates for Kähler manifolds with bounded Nash entropy

The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.

The article proves a new entropy formula for surfaces with boundaries.

We study the problem of existence of F-structures on compact complex surfaces, giving a complete classification modulo the gap in the classification of surfaces of class VII. We then use these results to study the minimal entropy problem for compact complex surfaces. For instance we prove that compact Kahler surfaces o…

Compact foliations preserve entropy if leaves are strictly convex projective.

Using the definition of entropy of a family of increasing distances on a compact metric set given in [10] we introduce a notion of Finsler entropy for smooth distributions and Stefan-Sussmann foliations. This concept generalizes most of classical topological entropy on a compact Riemannian manifold : the entropy of a f…

Ancient curve flows classified into specific types.

Study entropy bounds and finiteness for symmetric self-shrinkers.

Study on non-archimedean μ-entropy for toric varieties, proving existence and uniqueness.

The study shows a finite number of groups acting on hyperbolic spaces with bounded entropy and compact quotient.

In this paper, we develop a new approach to prove the $W$-entropy formula for the Witten Laplacian via warped product on Riemannian manifolds and give a natural geometric interpretation of a quantity appeared in the $W$-entropy formula. Then we prove the $W$-entropy formula for the Witten Laplacian on compact Riemannia…

Geodesic flows on compact manifolds without conjugate points are shown to have a unique measure of maximal entropy.

Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.

Let $f: Y \rightarrow X$ be a continuous map between a compact real analytic Kähler 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 of the diastatic entropy of $(X,g_0)$ and the degree of $f$. When the lower bound i…

The entropy of geodesic currents on hyperbolic surfaces is bounded by their self-intersection number.

We prove the compactness of self-shrinkers in $\mathbb R^3$ with bounded entropy and fixed genus. As a corollary, we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus.

This note relaxes conditions for Kähler metrics with bounded entropy and scalar curvature.

Study stability of compact Ricci solitons using entropy variations.

We survey several notions of entropy related to a compact manifold of negative curvature, some relations between them, and the rigidity problems.

In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and poten…

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+ε}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{hT}/T)$ simple clos…

We compute the second variation of the Ricci expander entropy and briefly discuss the linear stability of compact negative Einstein manifolds.

Harmonic manifolds of hypergeometric type have entropy bounds related to real hyperbolic spaces.

The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.

We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.

Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.

In this note we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to $S^1 \times S^{n-1}$ for each $n \geq 2$ in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely …

Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corol…

Compactness theorem for manifolds with scalar curvature and entropy bounds.

A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+α}$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nod…

We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and pa…

An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^\infty$ functions and has positive topological entropy is constructed.

Entropy measures geodesic flow complexity.

This paper controls a boundary term in Huisken's formula for entropy.

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface…

We show that the entropy of a finitely generated pseudogroup (resp., of a foliation of a compact Riemannian manifold) can be calculated by suitable counting separated pseudo-orbits (resp., pseudoleaves).

We find the entropy's infinite-size behavior in complex manifold sections.

Researchers set entropy limits for specific types of self-shrinkers.

We consider the entropy of the solution to the heat equation on a Riemannian manifold. When the manifold is compact, we provide two estimates on the rate of change of the entropy in terms of the lower bound on the Ricci curvature and the spectral gap respectively. Our explicit computation for the three dimensional hype…

The study proves compactness and structure of Ricci flow limits.

We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.

Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

We introduce the notion of a stationary random manifold and develop the basic entropy theory for it. Examples include manifolds admitting a compact quotient under isometries and generic leaves of a compact foliation. We prove that the entropy of an ergodic stationary random manifold is zero if and only if the manifold …

The study bounds entanglement entropy for coherent states on Kähler manifolds.

Extends finite entropy measures in Kähler geometry.

The study calculates Weyl entropy in spacetime regions and shows its monotonic behavior.