The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths.
method Introducing combinatorial Calabi flows and proving their long time existence and global convergence.
result Proves the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary.
Anosov magnetic flows on surfaces are characterized.
problem Characterizing Anosov magnetic flows on surfaces.
method Using Wojtkowski's quotient bundle, necessary and sufficient conditions are derived.
result Necessary and sufficient conditions for Anosov magnetic flows on surfaces are established.
The paper proves the existence of surfaces of section for geodesic flows on closed surfaces.
problem Existence of surfaces of section for geodesic flows on closed surfaces.
method Study of configurations of simple closed geodesics and use of the curve shortening flow.
result Construction of surfaces of section that intersect or have hyperbolic components in their boundary.
Study describes global existence and convergence of flows on surfaces and fibrations.
problem Global existence and convergence of flows on surfaces and fibrations.
method Complete description of Ricci-Yang-Mills flow and pluriclosed flow on Tk bundles over Riemann surfaces. result Equivalence of solutions to generalized Ricci flow and pluriclosed flow with symmetry.
Proves stability of geodesic flows on closed surfaces.
problem Stability of geodesic flows on closed surfaces.
method Generic Riemannian metrics and Reeb flows.
result Proves C2-stability conjecture for geodesic flows. 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 K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
The study derives formulas for functionals on surface with boundary under harmonic Ricci flow.
problem Understanding functionals along harmonic Ricci flow on surfaces with boundaries.
method Derivation of formulas for functionals under harmonic Ricci flow.
result Established formulas for functionals on surface with boundary.
Study models Ricci flow on complex surfaces, showing mixed behavior.
problem Understanding long-time behavior of Ricci flow on complex surfaces.
method BiLipschitz models for 4-manifolds (minimal surfaces of general type).
result Exhibits a combination of expanding and static behavior.
Willmore flow preserves low energy surfaces to planes.
problem Preserving low energy surfaces to planes under Willmore flow.
method Willmore flow equation for complete, properly immersed surfaces in Rn.
result Complete Willmore surfaces with low energy converge to planes.
Homoclinic orbits found in geodesic flows on surfaces.
problem Existence of homoclinic orbits in geodesic flows.
method Kupka-Smale metric on closed surfaces.
result Homoclinic orbits for all hyperbolic geodesics.
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
Fractional combinatorial flow improves surface conformal structures.
problem Improving discrete conformal structures on surfaces.
method Introducing a fractional combinatorial Calabi flow for discrete conformal structures on surfaces.
result Longtime existence and global convergence of the fractional combinatorial Calabi flow for various surface types.
Paper studies flow on hyperbolic surfaces to match boundary lengths.
problem Matching boundary lengths of hyperbolic surfaces.
method Combinatorial Yamabe flow on hyperbolic bordered surfaces.
result Flow converges exponentially to a surface with equal boundary lengths.
Study combinatorial Yamabe flow on infinite triangulated surfaces.
problem Solve discrete Yamabe problem on noncompact surfaces.
method Introduced and analyzed combinatorial Yamabe flow with short-time and long-time existence proofs.
result Established short-time and long-time existence of the flow, and proved convergence in hexagonal triangulations.
New flow generates surfaces with constant curvature.
problem Creating surfaces with specific curvature properties.
method Framed curvature flow, analyzing trajectory surfaces.
result Trajectory surfaces of constant mean or Gaussian curvature.
The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.
problem Finding hyperbolic bordered surfaces with prescribed boundary lengths.
method Fractional combinatorial Calabi flow and generalized combinatorial Yamabe flow.
result The flows converge to a hyperbolic surface with prescribed boundary lengths.
Article explains and implements mean curvature flow for surface parametrization.
problem Surface parametrization challenges.
method Conformalized mean curvature flow implementation.
result Demonstrates effectiveness of mean curvature flow for surface parametrization.
Geodesic flows on certain surfaces are shown to be semi-conjugate to expansive flows.
problem Understanding geodesic flows on compact surfaces without conjugate points.
method Time-preserving semi-conjugation to a continuous expansive flow.
result Geodesic flows on compact surfaces without conjugate points of genus > 1 have a unique measure of maximal entropy.
The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci …
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
Study curve shortening flows on specific surfaces, proving properties and existence.
problem Analyzing curve shortening flows on rotational surfaces with negative Gauss curvatures.
method Assume negative Gauss curvatures and conditions on Gauss curvature and curve curvature. Prove curve remains a graph and establish flow properties.
result Prove the curve remains a graph over parallels and establish long-time existence of the flow.
Proves multiplicity one conjecture for surface flows, with applications in flow regularity.
problem Proving multiplicity one for mean curvature flows of surfaces.
method Analyzing blow-up limits and using level set flow properties.
result Blow-up limits of mean curvature flows have multiplicity one.
Given an embedded cylinder in an arbitrary surface, we give a gauge theoretic definition of the associated Goldman flow, which is a circle action on a dense open subset of the moduli space of equivalence classes of flat SU(2)-connections over the surface. A cylinder in a compact nonorientable surface lifts to two cylin…
Study classifies ruled surfaces from mean curvature flow solutions.
problem Investigating surfaces with specific mean curvature equations.
method Classifying ruled and translation surfaces in Euclidean space.
result Ruled and translation surfaces are cylindrical.
The paper explores Kα-translators on parallel and canal surfaces in 3D space.
problem Investigating conditions for Kα-translators on parallel and canal surfaces. method Analyzing the conditions for Kα-translators on parallel surfaces and canal surfaces, proving their properties and existence. result No Kα-translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed w, while the rotational surface itself is a Kα-translator. Study curve shortening flow on Riemann surfaces with conic singularities.
problem Analyzing curve shortening flow on surfaces with conic singularities.
method Generalized Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. Reproofed Gage-Hamilton-Grayson theorem. Proved CSF can't touch conic singularities with cone angles ≤ π.
result CSF can't touch conic singularities with cone angles ≤ π for embedded simple closed curves.
Gradient flows for surface energies with tensor fields are derived and analyzed.
problem Deriving consistent gradient flows for surface energies involving tensor fields.
method Introducing different gauges of surface independence and demonstrating their effects on energy decrease.
result Consistent choice of gauge and time derivative is necessary for energy decrease.
The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.
problem Finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
method Introducing combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings.
result Proves longtime existence and global convergence of combinatorial curvature flows.
We define a parabolic flow of pluriclosed metrics. This flow is of the same family introduced by the authors in \cite{ST}. We study the relationship of the existence of the flow and associated static metrics topological information on the underlying complex manifold. Solutions to the static equation are automatically H…
Simple Ricci flow proof for Riemann surfaces.
problem Uniformization theorem of Riemann surfaces
method Ricci flow and Hamilton's isoperimetric estimate
result Simple proof of uniformization theorem
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
The paper studies how surfaces move by mean curvature flow and what happens at singular points.
problem Understanding the behavior of surfaces moving by mean curvature flow at singular points.
method Proves that tangent flows at singular times are smooth shrinkers, with a new local Gauss-Bonnet formula.
result Smooth shrinkers without branch points if the initial surface is embedded in 3-manifold.
Study shows generic surfaces avoid complex flow patterns.
problem Understanding flow patterns of surfaces in 3D space.
method Analyzes mean curvature flow of closed surfaces in R3. result Non-cylindrical self-shrinkers cannot arise generically.
Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curv…
Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.
problem Stability of surface diffusion and mean curvature flows in flat tori.
method Existence and convergence of flows starting close to stable critical sets, proven for all times.
result Flows converge exponentially fast to stable critical sets in flat tori.
The pluriclosed flow preserves Vaisman condition on compact complex surfaces.
problem Preserving Vaisman condition under pluriclosed flow.
method Pluriclosed flow on compact complex surfaces.
result Preserves Vaisman condition if and only if starting metric has constant scalar curvature.
Paper proves uniqueness of Ricci flows from nonatomic measures on surfaces.
problem Existence and uniqueness of Ricci flows from nonatomic Radon measures.
method Combining previous work, established existence and proved uniqueness.
result Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces.
Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
problem Noncompact cases of Gauss Curvature Flow on revolution surfaces.
method Examines two noncompact cases of Gauss Curvature Flow on revolution surfaces.
result Examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
We study the behaviour of the Ricci Yang-Mills flow for U(1) bundles on surfaces. We show that existence for the flow reduces to a bound on the isoperimetric constant. In the presence of such a bound, we show that on S2, if the bundle is nontrivial, the flow exists for all time. For higher genus surfaces the flow al…
In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a Kähler-Einstein surface. We prove that the blow-up flow Σs∞ at a singular point (X0,T0) of a symplectic mean curvature flow Σt or of a Lagrangian mean curvature flow Σt is …
Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the un…
Study shows unique tangent flow for Lagrangian surfaces with bounded mean curvature.
problem Understanding the behavior of Lagrangian surfaces with bounded mean curvature.
method Analyzing zero Maslov Lagrangian mean curvature flow in C2 with bounded mean curvature. result The tangent flow at a singular point is unique if the mean curvature stays uniformly bounded.
Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to …
The paper studies alternating links in thickened surfaces using flow lattices and disc mutations.
problem Understanding alternating links in thickened surfaces and their invariants.
method Using integer flows on Tait graphs and disc mutations, the paper proves invariants and compares link properties.
result Found alternating knots with isometric flow lattices but different linking forms.
In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that …
We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the cond…