Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
The article calculates the F-convergence rate for Ricci flows with closed and smooth tangent flows.
problem Analyzing the convergence rate of Ricci flows with specific tangent flows.
method Calculating the F-convergence rate for Ricci flows with closed and smooth tangent flows. result A Ricci flow with closed and smooth tangent flow is ∣logλ∣−θ close to its tangent flow in the F-sense. Smooth orbit equivalence proves metric equivalence for geodesic flows.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
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.
Establishes smooth Ricci flows from convex surfaces in 3D space.
problem Existence and uniqueness of Ricci flow starting from convex surfaces.
method Smooth Ricci flows starting from smooth convex surfaces.
result Uniform convergence of metrics to initial convex surface.
Study on when smooth Ricci flow remains smooth at the start.
problem When does a smooth Ricci flow remain smooth down to the initial time?
method Curvature estimates and lower Ricci bounds in three dimensions.
result Positive results for flows with lower curvature bounds, negative for others.
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
Paper studies smoothness of bi-conformal heat flow on 4-manifolds.
problem Smoothness of bi-conformal heat flow on 4-manifolds.
method Introduces bi-conformal heat flow (bi-CHF) and proves global smoothness without finite time singularities.
result Global smoothness and no finite time singularity for bi-conformal heat flow.
Turing complete flow on 4-sphere preserves volume.
problem Creating a Turing complete flow on a 4-sphere.
method Smooth, conservative flow on the 4-sphere.
result Achieved a Turing complete, volume-preserving flow.
Let {Tt} be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let μ be an ergodic measure of maximal entropy. We show that either {Tt} is Bernoulli, or {Tt} is isomorphic to the product of a Bernoulli flow and a rotational flow. Appli…
In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time t∈[0,∞) with unif…
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.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Researchers relax the CVF's smoothness requirement to create more flexible flow models.
problem Challenges in constructing flexible density models due to the CVF's smoothness requirement.
method Introduce L-diffeomorphisms as generalized transformations that may violate smoothness on zero Lebesgue-measure sets. result The relaxation allows for the use of non-smooth activation functions like ReLU in residual flows.
Solves Ricci flow on Riemann surfaces with measure initial data.
problem Existence and smoothness of Ricci flow on Riemann surfaces.
method Formulation and solution of existence problem using Ricci flow.
result New examples of nongradient expanding Ricci solitons.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
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.
In this note we prove that the level-set flow of the topologist's sine curve is a smooth closed curve. In previous work it was shown by the second author that under level-set flow, a locally-connected set in the plane evolves to be smooth, either as a curve or as a positive area region bounded by smooth curves. Here we…
Survey on preserving curvature bounds for non-smooth Ricci flow.
problem Preserving curvature bounds for non-smooth initial data in Ricci flow.
method Survey of various weak initial data and preservation of curvature bounds.
result Various curvature lower bounds preserved up to a constant for non-smooth initial data.
We introduce the conical Kähler-Ricci flow modified by a holomorphic vector field. We construct a long-time solution of the modified conical Kähler-Ricci flow as the limit of a sequence of smooth Kähler-Ricci flows.
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
Smooth flows for physical systems with smooth energies and forces.
problem Smooth energies for physical simulations and force computation.
method Smooth mixture transformations on compact intervals and hypertori, using root-finding and the inverse function theorem.
result Smooth flows allow training by force matching and use as molecular dynamics potentials.
We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metri…
FLUID uses flows to unify filtering and smoothing for complex systems.
problem Bayesian filtering and smoothing for high-dimensional nonlinear systems.
method FLUID encodes observation histories into a fixed summary statistic, using flows for filtering and smoothing.
result FLUID provides accurate approximations of filtering and smoothing distributions.
Ricci flow smooths locally collapsing manifolds with controlled curvature.
problem Locally collapsing manifolds with controlled Ricci curvature.
method Ricci flow for a definite period of time, detecting collapsing infranil fiber bundles.
result Topological conditions detect collapsing infranil fiber bundles.
Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when n=2, the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, th…
Let (Ft) be a smooth flow on a smooth manifold M and h:M→M be a smooth orbit preserving map. The following problem is studied: suppose that for every point z of M there exists a germ of a smooth function fz at z such that near z we have that h(x)=Ffz(x)(x). Can the functions (fz) be glued …
A flow defined by a nonsingular smooth vector field X on a closed manifold M is said to be parameter rigid if given any real valued smooth function f on M, there are a smooth funcion g and a constant c such that f=X(g)+c holds. We show that the parameter rigid flows on closed orientable 3-manifolds are sm…
Study vector fields and flows on singular spaces like submanifolds.
problem Understanding vector fields and flows on singular spaces.
method Integrate derivations of the C∞-ring of global smooth functions into flows. result Derivations integrate to smooth flows on subcartesian spaces.
Paper proves minimizing movements match smooth droplet flow in 3D.
problem Consistency of minimizing movements with smooth mean curvature flow.
method Proved minimizing movements coincide with smooth droplet flow.
result Minimizing movements and smooth mean curvature flow are consistent in 3D.
Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} that meet in triples at equal angles along smooth edges and higher order junctions on lower dimensional faces is a natural extension of classical mean curvature flow. We call such a flow a mean curvature flow with triple edges. We show that if a smoot…
For any n-dimensional smooth manifold Σ, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in Σ are cylindrical (of convex type) if the flow converges to a smooth hypersurface M∞ (maybe empty) at infinity. Previously this was shown (i) for n≤7,…
In this paper, we study a family of curves on S2 that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective pla…
The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.
problem Constructing Ricci flow solutions for non-smooth metrics in four dimensions.
method Ricci DeTurck flow on closed manifolds with initial values in W2,2. result Constructs solutions to Ricci flow for non-smooth metrics in four dimensions.
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.
Unified flow solves Lp Christoffel-Minkowski problem for p>1.
problem Solving the Lp Christoffel-Minkowski problem for p>1. method Anisotropic expanding flow of smooth hypersurfaces with speed ψσk(λ)α. result The flow converges to a solution of the Lp Christoffel-Minkowski problem. The (α,β)-Ricci-Yamabe flow exists on closed manifolds.
problem Existence of solutions to the (α,β)-Ricci-Yamabe flow. method Showed short time existence and established long time existence theorems.
result Existence of smooth solutions to the (α,β)-Ricci-Yamabe flow on closed manifolds. 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.
In this note, we show that the conical Kähler-Ricci flows introduced in \cite{CYW} exist for all time t∈[0,∞) in the weak sense. As a key ingredient of the proof, we show that a conical Kähler-Ricci flow is actually the limit of a sequence of smooth Kähler-Ricci flows.
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 prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space B21,p(M,Λ2) for p>4. The Donaldson geometric flow was introduced by Simon Dona…
Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.
problem Existence time of the Willmore flow for various initial conditions.
method Established minimal existence time for Willmore flow using geometric data and conservation laws.
result Minimal existence time is a function of geometric data for general weak Lipschitz initial data.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. New formulations for Ricci flows without smoothness.
problem Characterize Ricci flows without smooth solutions.
method Weak formulations of super Ricci flows with saturation condition.
result Generalized formulations for singular settings.
Study shows Brakke flow's non-triviality for smooth boundaries in codimension 1.
problem Understanding Brakke flow's non-triviality for smooth boundaries in codimension 1.
method Analyzing spacetime Brakke flow constructed by Buet et al. for initial varifolds.
result Support of mass measure of spacetime Brakke flow coincides with classical mean curvature flow's support.
Smooth convergence shown for curve diffusion flows.
problem Embeddedness and global existence of curves.
method Exponentially fast convergence established.
result Smooth convergence for curve diffusion flows.