Generic low-entropy hypersurfaces in 4-6D flow with only generic singularities.
problem Analyzing mean curvature flow of low-entropy hypersurfaces.
method Proving flow encounters only generic singularities for specific entropy conditions.
result Proves flow encounters only generic singularities for low-entropy initial data.
Low-entropy surfaces can be flowed into spheres and cylinders.
problem Proving mean curvature flow for low-entropy hypersurfaces.
method Low-entropy density drop argument and recent work on hypersurfaces.
result Closed hypersurfaces with entropy ≤ 2 can be flowed into spherical and cylindrical shapes.
We show that if Σ⊂R4 is a closed, connected hypersurface with entropy λ(Σ)≤λ(S2×R), then the level set flow of Σ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in R4 of low 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…
Gradient flow in softmax models tends to produce low-entropy outputs.
problem Understanding the training dynamics of softmax-based models.
method Analysis of gradient flow dynamics in the value-softmax model.
result Gradient flow drives optimization towards low-entropy solutions.
Curve shortening flow converges to a point with entropy bound.
problem Analyzing the behavior of curves under shortening flow near singularities.
method Analyzes blow-up limits and uses entropy bounds to prove convergence.
result Initial curves with entropy bound converge to a round point in finite time.
Entropy for submanifolds in hyperbolic space defined.
problem Entropy for submanifolds in hyperbolic space.
method Entropy defined analogous to Euclidean space.
result Entropy monotonicity along mean curvature flow in low dimensions.
We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.
In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than Λn−2. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption H−2⟨x,ν⟩>0 is possible and as an application we …
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.
In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in R3. Namely, if the flow has a spherical or cylindrical singularity at a space-time point X=(x,t), then there exists a positive ε=ε(X)>0 such that the flow is mean convex in a …
Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.
Backwards uniqueness proved for flows with asymptotically conical singularities.
problem Proving uniqueness of mean curvature flows with specific singularities.
method Developed new global tools to handle singularities, asymptotic structure, and smooth parts of flows.
result Backwards uniqueness for mean curvature flows with asymptotically conical singularities proved.
Study curve shortening flow in high dimensions with boundary constraints.
problem Understanding the behavior of curves in high-dimensional spaces with boundary conditions.
method Used curvature and higher-derivative estimates, Stahl-type maximum principle, and blow-up analysis.
result Flow converges to a shrinking semicircle model or has only semicircle boundary singularities in low entropy regimes.
Mathematical analysis of SNE and t-SNE for dimension reduction.
problem Optimal mapping of high-dimensional data to low dimensions.
method Gradient flow of relative entropy to minimize the distance between points.
result The diameter of the evolving sets remains bounded for SNE but may blow up for t-SNE.
Study proves existence and uniqueness of ancient flows from cones.
problem Existence and uniqueness of ancient rescaled mean curvature flows.
method Proved existence and uniqueness using strong uniqueness theorem.
result Proved existence and uniqueness of ancient flows from cones.
Density destructors simplify complex PDFs to maximize entropy, linking to information theory.
problem Complex multivariate PDFs are hard to analyze.
method Invertible transforms that progressively remove structure from PDFs.
result Density destructors can improve estimates of information theoretic quantities.
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.
Entropy measures geodesic flow complexity.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.
problem Smoothness of mean curvature flow for generic initial data.
method Long-time existence and uniqueness result for ancient mean curvature flows.
result Smooth mean curvature flow until disappearance in a round point for low-entropy hypersurfaces in 4D.
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
The paper proves the monotonicity of a modified Perelman's W-entropy for mean curvature flow.
problem Proving the monotonicity of Perelman's W-entropy for mean curvature flow.
method Modified definition of K. Ecker's W-entropy and Hamilton's Harnack inequality.
result The modified W-entropy is monotonically decreasing in time.
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
New method trains normalizing flows using entropy-regularized transport.
problem Training continuous normalizing flows efficiently.
method Formulates flows as gradients of scalar potentials, training only these potentials.
result Trains normalizing flows without explicit flow computation during training.
The article proves a new entropy formula for surfaces with boundaries.
problem Entropy formula for surfaces with boundaries.
method Established a monotonicity formula of Hamilton type entropy.
result Entropy functional and W-functional relation studied. Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.
problem Understanding entropy changes in flows near hyperbolic metrics.
method Analysis of geodesic flow on Riemannian manifolds with variable negative curvature.
result Topological entropy strictly decreases along normalized Ricci flow near hyperbolic metrics.
Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces.
problem Proving uniqueness of measure of maximal entropy for geodesic flows on surfaces.
method Analyzes geodesic flows on closed orientable C^∞ surfaces, proving uniqueness of measure of maximal entropy and at most one SRB measure.
result Proves uniqueness of measure of maximal entropy for geodesic flows on surfaces, covering previous results and new examples.
Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.
problem Entropy functional for Riemannian foliations and its relation to Ricci flow.
method Introduced entropy functional and related its gradient flow to transverse Ricci flow.
result Entropy functional is monotonic along transverse Ricci flow and related to it.
Ancient Ricci flows with nonnegative curvature operator have bounded entropy.
problem Conditions for bounded entropy in ancient Ricci flows.
method Used Perelman's entropy and Hamilton's trace Harnack inequality.
result Curvature operator nonnegativity is not necessary for bounded entropy.
Liouville entropy increases strictly along Ricci flow on surfaces.
problem Understanding the behavior of Liouville entropy under Ricci flow.
method New expression for Liouville entropy derivative, proof of positivity in specific directions.
result Liouville entropy is strictly increasing along normalized Ricci flow for 1/6-pinched metrics.
We study a flow of G2 structures which induce the same Riemannian metric which is the negative gradient flow of an energy functional. We prove Shi-type estimates for the torsion tensor along the flow. We show that at a finite-time singularity the torsion must blow-up, so the flow exists as long as the torsion remain…
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.
A new method for sampling high-dimensional distributions overcomes overfitting.
problem Overfitting in energy-based models during gradient descent.
method Mean-field microcanonical gradient descent, which samples multiple data points simultaneously.
result The method reduces entropy loss while maintaining likelihood fit, improving overfitting issues.
Ancient curve flows classified into specific types.
problem Classifying ancient finite-entropy curve shortening flows.
method Proving flow types through mathematical analysis.
result Ancient flows are one of several specific types.
Quantizes Kähler-Ricci flow for Fano manifolds.
problem Optimal degeneration for Fano manifolds.
method Geometric quantization of Kähler-Ricci flow and entropy functional.
result Established convergence to original flow and entropy.
In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eig…
In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ri…
In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow to the setting with free boundary. A major new challenge in the free boundary setting is to derive an a priori bound for the ratio between the norm of the second fundamental form and the mean curvature. We establish…
In this paper we aim to find a measure for the diversity of cash flows between agents in an economy. We argue that cash flows can be linked to probabilities of finding a currency unit in a given cash flow. We then use the information entropy as a natural measure of diversity. This leads to a hirarchical inequality meas…
In 1870s, L. Boltzmann proved the famous H-theorem for the Boltzmann equation in the kinetic theory of gas and gave the statistical interpretation of the thermodynamic entropy. In 2002, G. Perelman introduced the notion of W-entropy and proved the W-entropy formula for the Ricci flow. This plays a crucial role in…
In this note we determine the first two derivatives of the classical Boltzmann-Shannon entropy of the conjugate heat equation on general evolving manifolds. Based on the second derivative of the Boltzmann-Shannon entropy, we construct Perelman's F and W entropy in abstract geometric flows. Monotonicity of the entropies…
In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.
New flow method solves Christoffel-Minkowski problem.
problem Solving Christoffel-Minkowski problem.
method Entropy preserving curvature flow with global term.
result Entropy preserving flow solves Christoffel-Minkowski problem.
A new method for generative modeling of discrete data using geometric latent subspaces.
problem Learning generative models for discrete data with statistical dependencies.
method Geometric latent-subspace framework in exponential parameter space of product manifolds of categorical distributions.
result Low-dimensional latent space encodes statistical dependencies and accurately models high-dimensional discrete data.
New Finsler flow on 2-torus has chaotic dynamics.
problem Constructing chaotic dynamics on a 2-torus.
method Using Berger and Turaev's theorem, constructing a Finsler metric.
result Found a Finsler geodesic flow with positive metric entropy.