arXiv research

Study shows rigidity for entropy minimizers in non-monotone cases.

Entropy rigidity proven for 3D and higher convex projective manifolds.

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

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

Maximal representations show strong entropy rigidity.

The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

Entropy rigidity theorem for cusped Hitchin representations.

The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.

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

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

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.

New entropy functionals for curved spaces help predict shape behavior.

Research shows surfaces close to planes in Hausdorff distance.

We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as …

The paper connects currents and entropy in hyperbolic 3-manifolds.

In this note we give a short proof to the rigidity of volume entropy. The result says that for a closed manifold with Ricci curvature bounded from below, if the universal cover has maximal volume entropy, then it is the space form. This theorem was first proved by F. Ledrappier and X. Wang in [1].

Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.

In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric analog to the "entrop…

The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.

By means of a space-time Wasserstein control, we show the monotonicity of the W-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipatio…

For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to…

The paper develops techniques to study entropy and rigidity in RCD-spaces.

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 paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

The paper proves rigidity results for Einstein manifolds with specific geometric constraints.

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

The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.

Study shows only grim reaper cylinder for certain self-translating surfaces.

Study on rigidity of translating hypersurfaces not in graphical direction.

In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the ri…

We prove the following entropy-rigidity result in finite volume: if $X$ is a negatively curved manifold with curvature $-b^2\leq K_X \leq -1$, then $Ent_{top}(X) = n-1$ if and only if $X$ is hyperbolic. In particular, if $X$ has the same length spectrum of a hyperbolic manifold $X_0$, the it is isometric to $X_0$ (we a…

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend this result to $\mathsf{RCD}^{\ast}(-(N-1),N)$ spaces. While the upper bound is st…

Compact foliations preserve entropy if leaves are strictly convex projective.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

Entropy study on synthetic spaces with curvature bounds.

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…

Let f:(Y,g)->(X,g_0) be a non zero degree continuous map between compact Kähler manifolds of dimension greater or equal to 2, where g_0 has constant negative holomorphic sectional curvature. Adapting the Besson-Courtois-Gallot barycentre map techniques to the Kähler setting, we prove a gap theorem in terms of the degre…

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectr…

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy…

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.

Ancient Ricci flows are identified without curvature sign condition.

We study volume growth, entropy and stability for translating solitons of mean curvature flow. First, we prove that every complete properly immersed translator has at least linear volume growth. Then, by using Huisken's monotonicity formula, we compute the entropy of the grim reaper and the bowl solitons. We also give …

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuch…

We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and giv…