Paper studies heat flow for maps on manifolds, avoiding singularities.
problem Avoiding singularities in heat flow for maps on manifolds.
method Introduces regularized conformal heat flow for n-harmonic maps. result Regularized n-conformal heat flow does not develop finite time singularities. The paper proves a regularity theorem for Brakke flows near triple junctions.
problem Understanding the structure of triple junctions in Brakke flows.
method Establishes the ε-regularity theorem for k-dimensional Brakke flows near static, multiplicity-one triple junctions.
result The regular structure of triple junctions persists under weak mean curvature flow.
Improved sampling method using regularized Stein Variational Gradient Flow.
problem Improving the accuracy of sampling methods in machine learning.
method Proposed Regularized Stein Variational Gradient Flow to interpolate between SVGD and Wasserstein Gradient Flow.
result Established theoretical properties and provided preliminary numerical evidence of improved performance.
Proves an ε-regularity theorem for Ricci flows, leading to new singularity estimates.
problem Analyzing singularity models in Fano Kähler-Ricci flows.
method Proves ε-regularity theorem and uses it to derive new estimates.
result Establishes new estimates for singularity models of Fano Kähler-Ricci flows.
Study on the regularity of p-Gauss curvature flow near flat interfaces.
problem Regularity of p-Gauss curvature flow near flat interfaces. method Analysis of convex hypersurface near the interface.
result Regularity of the convex hypersurface near the interface.
Paper shows regularizing flow for conical Kähler-Ricci equations.
problem Regularizing property of conical Kähler-Ricci flow.
method Regularizing property of the twisted conical Kähler-Ricci flow from a positive closed current with zero Lelong number.
result Extends regularizing property to conical singularity case.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. 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.
Study proves a new flow method for mean curvature with volume change analysis.
problem Existence of BV flow via mean curvature flow.
method Elliptic regularization to prove existence of generalized BV flow.
result Proves existence of generalized BV flow with volume change expression.
In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.
The paper tackles safe reinforcement learning with convex regularization.
problem Safe reinforcement learning in complex, high-dimensional settings with safety constraints.
method Doubly-regularized RL framework combining reward and parameter regularization, formulated as a convex regularized objective with parametrized policies on an infinite-dimensional statistical manifold.
result Exponential convergence guarantees under sufficient regularization, robust theoretical insights and guarantees for safe RL.
Study on stability of network flow shrinkers with findings on instability of specific shapes.
problem Stability of regular shrinkers in network flow.
method Analysis of self-similarly shrinking solutions called regular shrinkers.
result All regular shrinkers with two or more enclosed regions can be perturbed away. Specific shapes like 4-ray star, 5-ray star, fish, and rocket are unstable among those with one enclosed region.
Survey discusses recent advances on Brakke flows and their properties.
problem Existence and regularity of Brakke flows.
method Analyzes recent progress including new theorems and proofs.
result Proof that branching singularities trigger dynamical instability.
Curve shortening flow's regularity depends on initial conditions after a certain time.
problem Understanding the regularity of evolving curves under curve shortening flow.
method Proposing and proving principles of controllable regularity based on initial conditions.
result No regularity estimate holds before a specific time, A/π. We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …
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.
In prior work the authors introduced a parabolic flow of pluriclosed metrics. Here we give improved regularity results for solutions to this equation. Furthermore, we exhibit this equation as the gradient flow of the lowest eigenvalue of a certain Schrödinger operator, and show the existence of an expanding entropy fun…
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. The abstract proves the existence and regularity of Brakke flows starting from a given set.
problem Existence and regularity of Brakke flows starting from a given set.
method Proves the existence and regularity of Brakke flows using a closed countably 1-rectifiable set in R^2.
result For almost all time, the flow locally consists of a finite number of embedded curves of class W^{2,2} whose endpoints meet at junctions with angles of 0, 60, or 120 degrees.
The paper proves smoothness of Brakke flows up to the end-time.
problem Smoothness of Brakke flows up to the end-time.
method Local regularity theorem for Brakke flows, extending White's theorem.
result Smooth extension of Brakke flows up to the end-time.
We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural p…
Proves regularity for Brakke flow near stationary half-plane.
problem Regularity of Brakke flow near stationary half-planes.
method Viscosity techniques from Savin applied to Brakke's flow.
result Proves C1,α regularity for flows close to stationary half-planes. The paper improves the regularity and existence of pseudo Calabi flow.
problem Improving the smoothness and existence of pseudo Calabi flow.
method Analyzing the initial conditions and using volume form closeness to smooth metrics.
result The pseudo Calabi flow becomes smooth immediately and exists for all time under certain conditions.
A new gradient flow for MMD with closed-form implementation.
problem Existing gradient flows either lack tractable numerical implementation or require strong assumptions.
method Introduces a (de)-regularized Maximum Mean Discrepancy (DrMMD) and its gradient flow.
result Guarantees near-global convergence for a broad class of targets in both continuous and discrete time.
Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.
problem Adjusting curvature flow to avoid a fixed region.
method Flow by powers of Gauss curvature, proving optimal curvature bounds and regularity.
result Proves optimal curvature bounds and long time existence for all dimensions and powers.
Study on Kähler-Ricci flow's Hölder regularity on compact manifolds.
problem Hölder regularity of Kähler-Ricci flow on compact Kähler manifolds.
method Adapting Hein-Tosatti's method for collapsing Calabi-Yau metrics, uniform spatial Hölder estimate obtained for all time.
result Uniform spatial Hölder estimate of Kähler-Ricci flow for all time.
Bi-Lipschitz flows approximate a wide range of distributions.
problem Characterizing the expressivity of bi-Lipschitz normalizing flows.
method Linking score regularity to transport map bi-Lipschitzness via probability flow ODE.
result Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are L1-dense among all probability densities. Study proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. Regularization leads to balancedness in deep linear networks.
problem Balancedness in deep linear networks.
method Geometric invariant theory and Riemannian geometry of fibers.
result Balancing flows converge to the balanced manifold at a uniform exponential rate.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures
Geodesic flow on submanifolds is shown to be Ck−1.
problem Regularity of geodesic flow on submanifolds.
method Analysis of Ck submanifolds with k≥2. result Geodesic flow and exponential map are Ck−1. Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.
problem Analyzing the inverse curvature flow of Legendre curves.
method Investigates the unique existence, monotonicity, and asymptotic behavior of the flow.
result The flow asymptotically converges to a self-similar solution, categorized by initial curve.
New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
We propose a potential flow generator with L2 optimal transport regularity, which can be easily integrated into a wide range of generative models including different versions of GANs and flow-based models. We show the correctness and robustness of the potential flow generator in several 2D problems, and illustrate t…
Regularizes f-divergences with MMD to analyze Wasserstein flows.
problem Limitations of f-divergences in measures' support. method Rewriting MMD regularization as Moreau envelope in RKHS, analyzing gradients.
result Analysis of Wasserstein flows of MMD-regularized f-divergences. We give the complete classification of regular projectively Anosov flows on closed three-dimensional manifolds. More precisely, we show that such a flow must be either an Anosov flow or decomposed into a finite union of T2×I-models. We also apply our method to rigidity problems of some group actions.
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.
We give complete classification of C^2-regular and non-degenerate projectively Anosov flows on three dimensional manifolds. More precisely, we prove that such a flow on a connected manifold must be either an Anosov flow or represented as a finite union of T2×[0,1]-models.
New estimate for Curve Shortening Flow improves graphical solutions.
problem Improving regularity estimates for Curve Shortening Flow.
method Generalizing delayed parabolic regularity for Curve Shortening Flow.
result Interior graphical estimate for Curve Shortening Flow.
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.
In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certai…
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kähler manifold. The goal of this paper is to give an ε-regularity theorem for the line bundle mea…
In this short note we announce a regularity theorem for Kähler-Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of Kähler-Ricci flow on Fano 3-manifolds. Moreover, we also present a partial C0 estimate to the Kähler-Ricci flow under …
Proves long-term smoothness of curved surfaces evolving under specific curvature rules.
problem Long-term regularity of curved surfaces evolving under p-Gauss curvature flow. method Transformed the curvature flow into a Monge-Ampère equation and studied its asymptotic cone.
result Proved regularity of the interface in all dimensions for $p> rac1n$.
New, shorter proofs for varifolds and flows with improved decay of flatness.
problem Proving regularity theorems for varifolds and flows with bounded first variation and forcing.
method Decay of flatness via weighted monotonicity formulas and viscosity approach.
result Improved proofs with decay of flatness and characterization of blow-ups.
Paper studies generic dynamics of MCFs with spherical singularities.
problem Characterizing the generic behavior of mean curvature flow with spherical singularities.
method Level set formulation of mean curvature flow, analysis of arrival time function.
result Generically, the arrival time function has at most C2 regularity.