Study smooth convergence of metric flows from F-limits.
problem Smooth convergence of F-limit flows. method Extensively studied metric flows and F-limits, showing smooth convergence at regular points. result Each regular point on the limit is a point of smooth convergence.
Ancient ovals are key blowup limits in 3D Ricci flow near singularities.
problem Understanding blowup limits in 3D Ricci flow near singularities.
method Proving ancient ovals are blowup limits if and only if spherical singularities accumulate.
result Ancient ovals are necessary and sufficient for blowup limits in 3D Ricci flow.
The paper examines the limit of harmonic flow on flat vector bundles.
problem Understanding the limiting behavior of harmonic flow on flat complex vector bundles.
method Analyzes the harmonic flow and proves the limit is isomorphic to a graded flat complex vector bundle.
result The limit of the harmonic flow on flat complex vector bundles is isomorphic to a graded flat complex vector bundle.
Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.
problem Geometric regularity of blow-up limits of the Kähler-Ricci flow.
method Established geometric regularity for Type I blow-up limits based on sequences of Ricci vertices.
result The limiting flow is continuous in time in Gromov-Hausdorff and Gromov-W1 distance. Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.
problem Analyzing mean curvature flow in high codimension Riemannian manifolds.
method Establishing codimension estimate, using quadratic pinching condition, gradient estimates.
result Existence of limiting flow in Euclidean space under cylindrical pinching condition.
Linear flows on inverse limits of tori are defined and it is shown that two linear flows on an inverse limit of tori are equivalent if and only if there is an automorphism of the inverse limit generating the equivalence.
Novel Ricci flow normalization for homogeneous spaces, focusing on flag manifolds.
problem Understanding the limiting behavior and symmetry properties of Ricci flow on homogeneous spaces.
method Introducing a novel normalization for the homogeneous Ricci flow and characterizing Gromov-Hausdorff limits.
result Full classification of Gromov-Hausdorff limits and detailed phase portraits for three-isotropy-summands flag manifolds.
Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.
problem Behavior of conical Kähler-Ricci flow as cone angle approaches zero.
method Analysis of limit behavior of conical Kähler-Ricci flow as cone angle tends to zero.
result Flow converges to a unique Kähler-Ricci flow with cusp singularity along the divisor.
The study proves compactness and structure of Ricci flow limits.
problem Understanding the structure of Ricci flow limits.
method Weak compactness theorem and structure theory development.
result Ricci flow limit spaces have a regular part with smooth convergence and a singular set of high codimension.
Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.
problem Analyzing the limits of Kähler-Ricci flow on Fano G-manifolds.
method Proves the Gromov-Hausdorff limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
result The limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
Characterizes limits of Ricci flows and their singularities.
problem Understanding the structure of non-collapsed limits of Ricci flows.
method Characterizes limits as smooth away from a set of high codimension, identifies tangent flows as gradient shrinking solitons, and stratifies singular set.
result Non-collapsed limits of Ricci flows are smooth away from a set of high codimension and have tangent flows as gradient shrinking solitons.
The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.
problem Analyzing limits of flows on Kähler surfaces and their convergence to solutions of equations.
method Using a property of limits of viscosity subsolutions.
result Proves convergence of flows to weak solutions of the Monge-Ampère equation.
Mirror flows converge to a limiting flow with a convex potential.
problem Incremental learning in mirror flows
method Rescaled trajectories converge to a limiting mirror flow
result Primal variable minimizes the loss over a time-dependent hypothesis set
Constructs approximate mean curvature flows for general varifolds.
problem Mean curvature flow for general initial data.
method Approximation of mean curvature flows using varifolds and iterated push-forwards.
result Approximate mean curvature flow converges to a spacetime Brakke flow under certain conditions.
Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.
problem Rigidity of invariant Ricci flow blowdown limits on nilpotent bundles with zero curvature.
method Construct a new functional to derive rigidity results for invariant Ricci flow blowdown limits on nilpotent principal bundles with zero associated curvature.
result Proves blowdown limit is locally an expanding Ricci soliton for three-dimensional Heisenberg group structure group.
It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean convex surfaces all limit flows are selfsimilar (static, shrinking or translating) if…
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power p for a smooth curvature function. result For 0<p≤1, limiting shape is always round as maximal existence time is approached. Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
The paper connects geodesic flows and limit sets on visibility manifolds.
problem Understanding dynamics and ergodic properties on non-compact visibility manifolds.
method Analyzing geodesic flows and Patterson-Sullivan measures on visibility manifolds without conjugate points.
result The positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow.
Study of t-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.
problem Investigating the t-Gauduchon Ricci-flat condition on non-Kähler manifolds. method Chern-Ricci flow approach, examples of non-Kähler Calabi-Yau manifolds, and geometric flow analysis.
result Examples of Chern-Ricci flow on non-Kähler Calabi-Yau manifolds that do not preserve the t-Gauduchon Ricci-flat condition. 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.
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the…
Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian on the uniqueness of the weak limit from sequential convergence constructio…
Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
Study collapsing Calabi-Yau metrics and flows on fiber spaces.
problem Understanding the behavior of Calabi-Yau metrics and flows during collapsing.
method Analyzing the collapsing of Calabi-Yau metrics and Kähler-Ricci flows on fiber spaces.
result Identify the collapsed Gromov-Hausdorff limit and bounds for Hausdorff measure.
We show that the twisted Kähler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilt…
We prove short time existence and uniqueness of the Laplacian flow starting at an arbitrary closed G2-structure. We establish long time existence and convergence of the Laplacian flow starting near a torsion-free G2-structure. We analyze the limit map of the Laplacian flow in relation to the moduli space of torsi…
Study shows superdiffusive behavior in geodesic flows on curved surfaces.
problem Understanding the statistical behavior of geodesic flows on curved surfaces.
method Proved nonstandard central limit theorem with superdiffusive normalisation (tlogt)1/2 for geodesic flows on nonpositively curved surfaces. result Geodesic flows exhibit superdiffusive behavior with correlations decaying at rate t−1. The paper shows translating solitons in R4 have SO(2) symmetry.
problem Understanding the symmetry of translating solitons in R4. method Analyzing the blow-up limits of embedded, mean convex mean curvature flow.
result Translating solitons in R4 have SO(2) symmetry. Study mean curvature flow of high codimension submanifolds in complex projective space.
problem Analyse mean curvature flow of high codimension submanifolds in complex projective space.
method Establish codimension estimate, prove convergence to smooth limiting flow, and prove decay estimate.
result Prove existence of limiting flow under cylindrical type pinching.
Study shows limits of certain normalizing flows in higher dimensions.
problem Understanding the representation power of normalizing flows in different dimensions.
method Rigorously established bounds on expressive power of basic normalizing flows.
result Limited representation power in higher dimensions, especially with moderate depth.
New method for geometric flows with surgery without smooth estimates.
problem Existence of geometric flows with surgery.
method Hybrid compactness theorem for weak limits.
result Existence of geometric flows with surgery in mean-convex surfaces.
The paper proves rigidity theorems for Type II singularities in Lagrangian flows.
problem Understanding Type II singularities in Lagrangian flows with zero Maslov class.
method Rigidity theorems for blow-up limits of Type II singularities.
result Generalized previous results from 2D to arbitrary dimensions.
The study examines gradient Ricci solitons with nonnegative curvature, proving properties of their blow-downs.
problem Characterizing gradient Ricci solitons with nonnegative curvature operator away from a compact set.
method Analyzing blow-downs and limits of Ricci flows to prove properties of solitons.
result No (n−1)-dimensional compact split limit Ricci flow can arise from the blow-down of (M,g). Gradient flows on graphons converge to curves on graphon space.
problem Optimizing functions on large, exchangeable graphs.
method Euclidean gradient flow on edge weights converges to a curve on graphon space.
result Gradient flows on graphons can be described as curves of maximal slope on graphon space.
The paper studies singular sets in Ricci flow limits, proving rectifiability and curvature bounds.
problem Understanding singular sets in Ricci flow limits.
method Stratification of singular sets, analysis of tangent flows, and geometric measure theory.
result Parabolic rectifiability of singular sets in certain dimensions and uniform curvature bounds.
Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.
problem Thom's gradient conjecture for nonlinear evolution equations.
method Extending and settling the conjecture in infinite dimensional problems using Łojasiewicz, L. Simon, and Kurdyka-Mostowski-Parusinski's foundational works.
result Uniqueness of the limiting direction and characterization of convergence rates for both classical and infinite dimensional settings.
Compactness theory for super Ricci flows provides convergence results.
problem Understanding convergence of super Ricci flows.
method Developed a compactness theory for super Ricci flows.
result Subsequential convergence to a metric flow under certain conditions.
Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
problem Fluctuations of boosting processes around their deterministic limit
method Stochastic perturbation analysis of ODEs in Banach spaces
result Rescaled deviations converge to a Gaussian process
Paper reconciles different Ricci flow approaches and proves weak solutions.
problem Proving weak solutions for Ricci flows with singularities.
method Introducing a novel hitting estimate for Brownian motion, compensating for lack of lower heat kernel bounds.
result Every noncollapsed limit of Ricci flows and singular Ricci flows are weak solutions.
We investigate the behavior of limit order books on the meso-scale motivated by order execution scheduling algorithms. To do so we carry out empirical analysis of the order flows from market and limit order submissions, aggregated from tick-by-tick data via volume-based bucketing, as well as various LOB depth and shape…
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.
problem Understanding the asymptotic geometry of ancient Ricci flows with nonnegative Ricci curvature.
method Analyze tangent flows at infinity and use estimates for noncollapsed F-limit metric solitons.
result Two dichotomy theorems for ancient Ricci flows: either the asymptotic volume ratio is zero or every tangent flow is a Ricci flat cone.
Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.
problem Analyzing convergence of Yang-Mills-Higgs flow for twisted Higgs pairs.
method Proves convergence to a reflexive twisted Higgs sheaf outside a closed subset.
result Limiting twisted Higgs sheaf is isomorphic to the double dual of graded twisted Higgs sheaves.
Ancient Ricci flows with bounded girth found in 3D and higher.
problem Finding ancient Ricci flows with bounded girth in dimensions 3 and higher.
method Invariant conditions on curvature and its derivatives under O(2)imesO(n−1) symmetry, proving Ricci flow invariance. result Construction of new ancient Ricci flows with positive curvature operator and bounded girth.
We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow.
Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analysis of tangent flows and blowup limits of special Lagrangian cones.
result Uniqueness of tangent flows in dimension two and any dimension when the link is connected.
We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>n+21 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…