We discuss recent results on minimal surfaces and mean curvature flow, focusing on the classification and structure of embedded minimal surfaces and the stable singularities of mean curvature flow. This article is dedicated to Rick Schoen.
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48 results for “minimal surface flow”
New surfaces show horocyclic flow isn't always minimal.
problem Complex dynamics on infinite fineness surfaces.
method Construction of infinite hyperbolic surfaces.
result Horocyclic flow is not minimal on infinite fineness surfaces.
Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.
problem Classifying minimal surfaces and solitons in hyperbolic 3-space.
method Investigates minimal surfaces and solitons to the mean curvature flow in hyperbolic three-space using specific product forms of curves.
result Provides classification results for minimal surfaces, hyperbolic translators, and conformal solitons.
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.
We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries…
The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.
problem Finding global solutions and convergence of minimal surface flow and translating solitons.
method Evolved surfaces over convex planar domains evolving by minimal surface flow, proving a priori estimates for translating solitons.
result Global solutions of minimal surface flow converge to translating solitons under suitable conditions.
New examples of mean curvature flow converge to minimal surfaces with multiplicity 2.
problem Constructing mean curvature flow examples in closed manifolds.
method Constructing new examples of mean curvature flow with convergence to minimal surfaces with multiplicity 2.
result Mean curvature flow examples converge to minimal surfaces with multiplicity 2.
The study examines the long-term behavior of mean curvature flows in closed 3-manifolds.
problem Understanding the long-term behavior of mean curvature flows in closed 3-manifolds.
method The approach involves constructing piecewise almost regular flows and applying perturbative arguments.
result The study constructs minimal surfaces in 3-manifolds via parabolic methods.
We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no…
The horocyclic flow on geometrically infinite surfaces shows recurrent irregular orbits or non-minimal closures.
problem Complex dynamics of horocyclic flow on geometrically infinite surfaces.
method Analyzing the recurrence and minimality of irregular orbits.
result Irregular orbits are recurrent or have non- minimal closures.
Boundary Singularities in Mean Curvature Flow and Total Curvature of Minimal Surface Boundariesmath.DG
The paper studies singularities in mean curvature flow and bounds on minimal surface total curvature.
problem Understanding singularities in mean curvature flow and bounds on minimal surface total curvature.
method Mean curvature flow and geometric analysis.
result The largest number k for which a minimal surface with total curvature less than k is a disk is greater than 3π.
Proves existence of mean curvature flow with surgery for free boundary surfaces.
problem Existence of mean curvature flow with free boundary.
method New approach for flows with surgery in free boundary setting.
result Flow converges to stable minimal surfaces without surgeries for large t.
We study the geodesic flow on the normal line congruence of a minimal surface in induced by the neutral Kähler metric on the space of oriented lines. The metric is lorentz with isolated degenerate points and the flow is shown to be completely integrable. In addition, we give a new holomorphic description …
New approach to heat flow for half-harmonic maps, related to minimal surfaces.
problem Heat flow for half-harmonic maps from to closed target manifolds.
method Classical approach using Dirichlet-to-Neumann operator for the Laplace equation.
result Analogous results to 1985 harmonic map heat flow, valid for finite-energy data.
Extends Newton's minimal resistance problem to Riemannian surfaces.
problem Minimal resistance on Riemannian surfaces.
method Derive resistance functional, analyze constrained minimization.
result Smooth extremals are loxodromes, global minimizers characterized.
Flow maps into minimal surfaces with free boundary.
problem Mapping surfaces into minimal submanifolds with free boundary.
method Combining Plateau-flow and Teichmüller harmonic flow to achieve half-harmonic maps.
result Flow produces a branched minimal immersion as time tends to infinity.
Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.
problem Proving convergence of Chern-Ricci flow on complex minimal surfaces.
method Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence; surface torsion estimate, uniform total variation bound, Green-weighted L^2 estimate, linear iteration of real Poisson equations.
result Uniform diameter estimates, volume non-collapsing estimates, Gromov-Hausdorff convergence for normalized Chern-Ricci flow on complex minimal surfaces.
We prove existence, uniqueness and convergence of solutions of the degenerate J-flow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on minimal surfaces of general type.
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.
problem Existence and properties of foliations in hyperbolic 3-manifolds.
method Mean curvature flow, surgery, min-max theory, foliations, continuity of minimal surfaces.
result Existence of smooth entire foliations in quasi-Fuchsian and hyperbolic 3-manifolds.
New approach to solving minimal surface system Dirichlet problem on smooth domains.
problem Solving Dirichlet problem for minimal surface system on smooth domains.
method Using mean curvature flow (MCF) with boundary conditions and non-negative Ricci curvature assumption.
result Existence of long-time mean curvature flow and existence result for exterior Dirichlet problem.
An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by…
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…
The study examines singularities in flows with curvature bounds and identifies unique tangent flows.
problem Analyzing singularities in mean curvature flows with curvature bounds.
method Examines tangent flows and uses stationary and area-minimizing cones to identify unique flows.
result For flows with , the tangent flow is unique when and is a regular cone.
Paper proves uniqueness of weak solutions for Plateau flow.
problem Proving uniqueness of weak solutions for Plateau flow.
method Used natural energy condition and alternative methods from Struwe.
result Proves uniqueness of weak solutions under natural condition.
Ricci flow stabilizes hyperbolic 3-manifolds near the hyperbolic metric.
problem Stability of Ricci flow on hyperbolic 3-manifolds.
method Normalized Ricci-DeTurck flow with exponential convergence to the hyperbolic metric.
result Normalized Ricci-DeTurck flow converges exponentially to the hyperbolic metric.
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 at a singular point of a symplectic mean curvature flow or of a Lagrangian mean curvature flow is …
A neural flow method minimizes Willmore energy for 2-surfaces in 3D space.
problem Minimizing Willmore energy for closed oriented 2-surfaces in 3D space.
method Introducing neural Willmore flow to model and minimize the Willmore energy using neural architectures.
result The neural flow reproduces expected round sphere and Clifford torus for genus 0 and 1 surfaces, respectively, and finds minimal Willmore surfaces for genus 2.
Minimal surfaces and average area ratio found to be maximized by hyperbolic metrics.
problem Finding sharp relations between minimal surface entropy and average area ratio.
method Ricci flow with surgery and invariant measures.
result Minimal surface entropy maximized by hyperbolic metrics among metrics with scalar curvature ≥ -6.
The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, …
In this paper, we study an -flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the -flows is a weak solution to the harmonic map flow. By an application of the -flow, we present a simple proof of an energy identity of a minimizing sequence in each homotopy class.
Distance between evolving hypersurfaces is a PDE solution.
problem Tracking the distance between evolving hypersurfaces.
method Elliptic and parabolic PDEs, mean curvature flow.
result Local Harnack inequalities for the distance between evolving hypersurfaces.
Minimal surface entropy and applications of Ricci flow on finite-volume hyperbolic 3-manifoldsmath.DG
The paper studies minimal surface entropy on hyperbolic 3-manifolds and compares it to the hyperbolic case.
problem Minimal surface entropy on hyperbolic 3-manifolds and its comparison to the hyperbolic case.
method Analysis of Ricci flow convergence and comparison of metrics with sectional and scalar curvature constraints.
result The entropy is maximized at the hyperbolic metric under certain curvature conditions.
Study on stability of hyperkähler flow in 4-manifolds.
problem Stability of hyperkähler flow in 4-manifolds.
method Extending results from mean curvature flow for minimal surfaces to hyperkähler flow.
result Obtained a dynamic stability theorem for hyperkähler flow.
We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the si…
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 …
Study on droplet flow on uneven surfaces, proving existence and properties.
problem Understanding droplet movement on irregular surfaces.
method Existence of smooth flow and 1/2-Hölder continuous minimizing movement solutions.
result Properties of minimizing movements including comparison principles and uniform boundedness.
We give a classification of compact solitons for the pluriclosed flow on complex surfaces. First, by exploiting results from the Kodaira classification of surfaces, we show that the complex surface underlying a soliton must be Kähler except for the possibility of steady solitons on minimal Hopf surfaces. Then, we const…
Unstable minimal surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for minimal surfaces in the Euclidean spacce was presented in \cite{s3}. We…
We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüll…
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a K…
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.
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
In this paper we introduce a flow on the spectral data for symmetric CMC surfaces in the -sphere. The flow is designed in such a way that it changes the topology but fixes the intrinsic (metric) and certain extrinsic (periods) closing conditions of the CMC surfaces. For rational times we obtain closed (possibly bran…
The study finds bounds on minimal surfaces in hyperbolic 3-manifolds.
problem Finding bounds on areas of stable minimal surfaces in hyperbolic 3-manifolds.
method Ricci flow with surgery to find lower bounds on minimal surface areas.
result The hyperbolic metric maximizes a functional defined by asymptotic counts of minimal surfaces.
Alternative solvability criterion for minimal surface equations and mean curvature flow.
problem Solvability of Dirichlet problem for minimal surface equation in non-mean convex domains.
method Introduces a structural condition from a second-order ODE to construct boundary barriers, applicable to unbounded domains and Hadamard manifolds.
result Allows solvability under geometric hypotheses different from classical Jenkins-Serrin theory, applicable to Euclidean space and mean curvature flow.
We give a survey of various existence results for minimal Lagrangian graphs. We also discuss the mean curvature flow for Lagrangian graphs.
The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preser…