Study shows rigidity for entropy minimizers in non-monotone cases.
problem Rigidity of entropy minimizers in non-monotone settings.
method Elementary proofs in non-monotone situations.
result Showed rigidity for minimizers of generalized Colding-Minicozzi entropies.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.
problem Understanding entropy in higher-codimension mean curvature flow.
method Measure-theoretical techniques and rigidity results for self-shrinkers.
result Existence of entropy minimizers and improved rigidity results.
Entropy rigidity shows volume bounds for convex projective manifolds.
problem Entropy rigidity of convex projective manifolds.
method Adapting Besson--Courtois--Gallot's entropy rigidity result to Hilbert geometries.
result Hilbert volume is bounded below by a constant depending only on dimension.
Surveying entropies for negatively curved manifolds and their relations.
problem Understanding entropy in negatively curved manifolds.
method Exploring various entropy concepts and their interconnections.
result Relations between different entropy notions for negatively curved manifolds.
Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.
problem Quantifying rigidity in Alexandrov spaces with curvature constraints.
method Using Gromov-Hausdorff distance and properties of Alexandrov spaces.
result Alexandrov spaces with curvature bounds are close to hyperbolic manifolds.
Maximal representations show strong entropy rigidity.
problem Entropy rigidity for maximal representations.
method Measurable hypertransversality, Gromov product, Bowen-Margulis-Sullivan measure.
result Strong entropy rigidity proved for maximal representations.
Monotonicity and rigidity of W-entropy proved in singular spaces.
problem Entropy behavior in singular metric measure spaces.
method Space-time Wasserstein control to show monotonicity and rigidity.
result Entropy dissipation rate and rigidity models in singular spaces.
Entropy rigidity theorem for negatively curved finite volume manifolds.
problem Proving entropy rigidity for negatively curved manifolds of finite volume.
method Entropy rigidity result in finite volume, comparing with compact case.
result Isometry of manifolds with same length spectrum.
The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.
problem Understanding the rigidity of Patterson-Sullivan systems and their applications.
method Generalization of Tukia's measurable boundary rigidity theorem for Patterson-Sullivan systems.
result Entropy rigidity for Anosov groups with Lipschitz limit sets.
Maximal volume entropy rigidity extended to RCD spaces.
problem Volume entropy rigidity for RCD spaces with bounded Ricci curvature.
method Extending known results to RCD spaces, dealing with lack of smooth structure.
result Rigidity result for compact RCD spaces with specific curvature bounds.
Entropy rigidity theorem for cusped Hitchin representations.
problem Entropy rigidity for Hitchin representations of cusped groups.
method Introduction of (1,1,2)-hypertransverse groups and transverse representations.
result Hausdorff dimension of conical limit set agrees with simple root entropy.
Entropy rigidity for Finsler flows but collapse for Reeb flows.
problem Entropy behavior of Reeb and Finsler flows on contact manifolds.
method Analysis of topological entropy for Reeb and Finsler flows.
result Uniform positive lower bound for Finsler flows but arbitrarily small topological entropy for Reeb flows.
The paper characterizes curvature-dimension conditions and related inequalities on Riemannian manifolds.
problem Curvature-dimension conditions and related inequalities on Riemannian manifolds.
method Information-theoretic approach to study curvature-dimension condition, rigidity theorems, and entropy differential inequalities.
result Equivalence of curvature-dimension condition and entropy differential inequalities on Riemannian manifolds.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
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.
problem Understanding entropy behavior in curved spaces.
method Introduced new entropy functionals for submanifolds of Cartan-Hadamard manifolds.
result Obtained sharp lower bounds on these entropies for certain closed hypersurfaces and observed a novel rigidity phenomenon.
Research shows surfaces close to planes in Hausdorff distance.
problem Understanding submanifolds with entropy close to one.
method Analyzing entropy and Hausdorff distance to prove rigidity.
result Submanifolds with entropy close to one are close to planes.
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.
problem Understanding the entropy of negatively curved 3-manifolds.
method Intersection of geodesic and conformal currents, proving sharp bounds.
result New proofs of Liouville entropy, minimal surface entropy, and Mostow Rigidity Theorem.
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].
The paper proves entropy power properties on Riemannian manifolds and Ricci flows.
problem Entropy power on Riemannian manifolds and Ricci flows.
method Proving concavity and convexity of Shannon entropy power for heat and conjugate heat equations on Riemannian manifolds and Ricci flows.
result Entropy power rigidity models on Einstein or quasi Einstein manifolds and shrinking Ricci solitons.
The study proves rigidity results for Alexandrov spaces with specific curvature and entropy conditions.
problem Proving rigidity results for Alexandrov spaces with curvature and entropy constraints.
method Analyzing Alexandrov spaces with curvature and entropy conditions, proving rigidity results.
result Universal covers of Alexandrov spaces with specific curvature and entropy conditions are isometric to hyperbolic spaces.
Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.
problem Positive entropy actions by higher-rank lattices in Lie groups.
method Analysis of sub-actions, fiber entropy upper semicontinuity, and conjugacy arguments.
result Actions by higher-rank lattices in SL(n,R) are conjugate to affine actions on (infra-)tori. 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.
problem Volume entropy rigidity for manifolds with lower integral Ricci curvature bound.
method Analyzing manifolds with specific integral Ricci curvature bounds, diameter, and volume entropy.
result The universal cover of the manifold is close to a hyperbolic space form under certain conditions.
The paper develops techniques to study entropy and rigidity in RCD-spaces.
problem Entropy and rigidity in RCD-spaces.
method Develops the barycenter technique for RCD-spaces and applies it to show entropy-volume inequalities.
result RCD-spaces with equality in entropy-volume inequality are locally symmetric.
Let f:Y→X be a continuous map between a compact real analytic Kähler manifold (Y,g) and a compact complex {hyperbolic manifold} (X,g0). In this paper we give a lower bound of the diastatic entropy of (Y,g) in terms of the diastatic entropy of (X,g0) and the degree of f. When the lower bound i…
The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.
problem Understanding the structure and behavior of Ricci shrinkers.
method Proving rigidity and ε-regularity theorems for Ricci shrinkers using entropy and curvature.
result Non-compact Ricci shrinkers are asymptotic to cones under certain curvature conditions.
The paper proves rigidity results for Einstein manifolds with specific geometric constraints.
problem Understanding the rigidity of Einstein manifolds under bounded covering geometry.
method Analyzing Einstein manifolds with bounded covering geometry to prove rigidity results.
result Compact Einstein manifolds with specific geometric properties are isometric to space forms.
The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.
problem Investigating Q-curvature and volume entropy on conformally flat manifolds.
method Introducing a new volume entropy, establishing identities, and proving rigidity results.
result Each polynomial growth polyharmonic function on such manifolds is of finite dimension, and the Cohn-Vossen inequality achieves equality under specific conditions.
Study shows only grim reaper cylinder for certain self-translating surfaces.
problem Characterizing self-translating surfaces in 3D space.
method Used parabolicity in a weighted setting and universally L-superharmonic functions.
result Characterized the grim reaper cylinder as the only finite entropy self-translating 2-surface in R^3 of width π and bounded from below.
Study on rigidity of translating hypersurfaces not in graphical direction.
problem Rigidity of translating hypersurfaces not in graphical direction.
method Proved rigidity results for complete graphical translating hypersurfaces under specific conditions.
result Entire graphical translating surfaces are flat under certain conditions.
The Clifford torus is unstable but rigid in mean curvature flow.
problem Stability and rigidity of the Clifford torus in mean curvature flow.
method Analysis of higher order phenomena, including entropy minimisation and infinitesimal deformations.
result The Clifford torus is locally unique as a self-shrinker for mean curvature flow.
The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.
problem Geometric obstructions and volume bounds in singular spaces.
method Developed new degree theorem for Alexandrov spaces using integral currents.
result Entropy-volume minimization prevents metric singularities in Gromov-Hausdorff limits.
Compact foliations preserve entropy if leaves are strictly convex projective.
problem Entropy rigidity for foliations by strictly convex projective manifolds.
method Analysis of foliated volume entropies and homeomorphisms.
result Equality in foliated volume entropies implies homothetic leaves.
The study constructs pressure form on Margulis spacetimes and proves their infinitesimal rigidity.
problem Understanding the infinitesimal rigidity of Margulis spacetimes.
method Constructing pressure form and studying its properties on the moduli space of Margulis spacetimes.
result Margulis spacetimes are infinitesimally determined by their marked Margulis invariant spectra.
Abstract: Study continuous maps between Kähler manifolds, proving a gap theorem.
problem Continuous maps between Kähler manifolds with negative curvature.
method Adapting Besson-Courtois-Gallot techniques to Kähler setting, proving gap theorem.
result Proves a gap theorem in terms of degree and diastatic entropies.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
Entropy study on synthetic spaces with curvature bounds.
problem Entropy functional on synthetic spaces with curvature bounds.
method Rigorous justification of entropy formula, monotonicity, and rigidity properties; heat kernel bounds.
result Bounds for heat equation solutions on synthetic spaces.
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…
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.
problem Understanding the growth rate and asymptotic behavior of k-surfaces in negatively curved 3-manifolds. method Proved results on the asymptotic behavior of high energy k-surfaces, including upper bounds and rigidity theorems. result Determined a rigid upper bound for the growth rate of quasi-Fuchsian k-surfaces in negatively curved 3-manifolds. The paper proves properties of Renyi entropy power on Riemannian manifolds.
problem Properties of Renyi entropy power on Riemannian manifolds.
method Proof of concavity, rigidity models, Aronson-Benilan estimates, NIW formula, entropy isoperimetric inequality.
result Rigidity models and intrinsic relationships for Renyi entropy power.
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…
Paper improves volume comparisons and entropy estimates for integral Ricci curvature.
problem Volume comparisons and entropy estimates for integral Ricci curvature.
method Several volume comparisons and an estimate for volume entropy.
result Improved estimates for volume entropy and algebraic entropy.
We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.