Classifies ancient convex curves in convex domains.
problem Ancient convex curve shortening flows on convex domains.
method Classification of convex ancient solutions.
result Ancient convex curves in convex domains classified.
Convex curves evolve into circles over time.
problem Deforming convex curves into circles.
method Generalized length-preserving flow for convex curves.
result Convex curves evolve into circles over time.
Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.
problem Mean curvature flow with surgery for compact mean convex hypersurfaces.
method Topological surgeries performed by the flow itself through nondegenerate cylindrical singularities, adjusted at smooth times.
result Extends previous results for 2-convex flows and constructs a flow for compact mean convex hypersurfaces.
Classifies ancient flows in a disc with boundary.
problem Ancient convex flows in a disc with boundary.
method Classifies flows using curve shortening.
result Ancient convex flows in a disc are classified.
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.
The study characterizes and rules out collapsing in convex ancient mean curvature flow.
problem Characterizing and ruling out collapsing in convex ancient mean curvature flow.
method Characterization and counterexamples.
result Collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane.
Unique ancient convex flow in a ball with free boundary found.
problem Classifying convex ancient free boundary mean curvature flows in the ball.
method Proof of existence and uniqueness in every dimension.
result A unique (modulo rotations and translations) convex ancient mean curvature flow found.
Flow deforms locally convex curves to curves of constant k-order width.
problem Evolve locally convex curves to curves of constant k-order width.
method Introduced a nonlocal curvature flow to evolve locally convex curves in the plane.
result The flow converges to a smooth, locally convex curve of constant k-order width as time goes to infinity.
Proves flows of two-convex Lagrangians are regular, global, and converge.
problem Proves regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case.
method Uses a newly discovered monotone quantity to control two-convexity.
result Proves results for the mean curvature flow of area-decreasing Lagrangian submanifolds.
Ancient convex solutions to flow equations are limited to simple shapes.
problem Characterizing ancient convex solutions to flow equations.
method Analyzing mean curvature flow and curvature functions of convex hypersurfaces.
result Ancient convex solutions to flow equations are limited to spherical, cylindrical, or planar shapes.
New global section found for geodesic flows on convex hypersurfaces.
problem Finding global sections for geodesic flows on convex hypersurfaces.
method Constructing a global hypersurface of section with an isometric involution.
result Generalized Birkhoff annulus to higher dimensions.
New Harnack inequality for curve shortening flow without convexity.
problem Proving a Harnack inequality for curve shortening flow without convexity.
method Developed a new Harnack inequality for one-dimensional mean curvature flow (curve shortening flow) that doesn't require convexity.
result Explicit time by which an initial curve becomes graphical under curve shortening flow.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. Geodesic flow mixing on convex projective manifolds proven.
problem Understanding mixing properties of geodesic flow on convex projective manifolds.
method Introduced biproximal unit tangent bundle and proved mixing properties.
result Geodesic flow is topologically mixing on biproximal unit tangent bundle.
The paper studies stability and singularities of a two-convex level set flow.
problem Stability and singularities of a two-convex level set flow.
method Assumes two-convex initial hypersurface and finitely many singular times, then shows the singular set has finitely many connected components.
result Near each connected component of the singular set, the perturbed flow has the same type of singular set.
New convex ancient solutions found for flows by high powers of curvature.
problem Existence of closed convex ancient solutions to curvature flows.
method Proves existence of closed convex ancient solutions with specific curvature flow speeds.
result Existence of non-homothetic convex ancient solutions for flows by high powers of curvature.
Flow deforms locally convex curves into target curves.
problem Deforming locally convex curves to target curves with same elastic energy.
method Curvature flow with nonlocal term to evolve curves.
result Flow deforms curves to target curves if elastic energies match.
Ancient Lagrangian flows get limited convex solutions.
problem Controlling convex solutions of Lagrangian flows at antiquity.
method Proving a Liouville type theorem with quadratic growth restrictions.
result Ancient convex solutions are unique.
We consider the mean curvature flow of compact convex surfaces in Euclidean 3-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite…
Two flows for convex curves converge to circles smoothly.
problem Curvature flow of convex closed plane curves.
method 1/κ^n type curvature flows for closed convex planar curves.
result Closed convex curves converge to a circle smoothly.
The paper studies curve shortening flows on non-convex surfaces.
problem Behavior of curve shortening flows on non-convex surfaces.
method Defined a graph property and proved its preservation under curve shortening flow.
result The curve becomes a graph after a finite time under the curve shortening flow.
Paper finds convexity in translating solitons for concave flows.
problem Understanding convexity in translating solitons for concave extrinsic flows.
method Analyzes convexity estimates for translating solitons evolving under concave functions in Rn+1. result Establishes convexity estimates for translating solitons of concave extrinsic geometric flows.
The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.
problem Deforming convex hypersurfaces in Euclidean space.
method Fully nonlinear curvature flow involving k-th elementary symmetric function and support function.
result Long-time existence and convergence of the flow under certain assumptions.
New ancient solutions found for curvature flow in 2D.
problem Ancient solutions for curvature flow in 2D.
method Constructing and classifying convex ancient solutions.
result All convex ancient solutions classified for α∈(32,1). Gradient flow converges to a minimal convex structure.
problem Finding the minimal convex structure in hyperbolic manifolds.
method Weil-Petersson gradient vector field of renormalized volume.
result The flow converges to the structure with minimum convex core volume.
Geometric inequalities for static convex domains in hyperbolic space proved.
problem Proving geometric inequalities for static convex domains in hyperbolic space.
method Using static convexity of flow hypersurfaces, new inequalities are derived.
result New family of geometric inequalities for static convex domains in hyperbolic space.
High codimension submanifolds evolve to convex shapes, leading to smooth limiting flows.
problem Evolution of high codimension submanifolds in Rn+k. method Proving asymptotic convexity and using it to show convergence to a smooth limiting flow.
result High codimension submanifolds evolve to convex shapes, and at singular times, rescaling converges to a smooth limiting flow.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
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
The paper studies a flow for convex capillary hypersurfaces in a ball, proving smooth convergence to a spherical cap.
problem Analyzing the behavior of convex capillary hypersurfaces under mean curvature flow.
method Introduced mean curvature flow for hypersurfaces in the unit Euclidean ball with capillary boundary. Proved smooth convergence to a spherical cap for strictly convex initial hypersurfaces.
result The flow preserves strict convexity and converges smoothly to a spherical cap for all positive time.
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.
problem Volume preserving Gauss curvature flow of convex hypersurfaces in hyperbolic space.
method Volume preserving flow with speed given by Gauss curvature power α, using Alexandrov reflection and hyperbolic curvature measures.
result Smooth solution remains convex and converges to a geodesic sphere exponentially.
New proof shows symmetry for certain curved surfaces in higher dimensions.
problem Understanding symmetries in curved surfaces evolving over time.
method Generalized previous result to higher dimensions, proving symmetry condition.
result Uniformly 3-convex translating solitons must have SO(n−1) symmetry. Proves planarity and convexity for ancient solutions of mean curvature flow.
problem Ancient solutions of mean curvature flow in higher codimension.
method Parabolically scale-invariant variation of planarity estimate, convexity proof for pinched solutions.
result Characterizes certain pinched complete ancient solutions and shrinkers in higher codimension.
The study classifies flows of finite curvature in 3D space.
problem Classifying flows of finite curvature in 3D space.
method Partial classification of eternal mean convex flows.
result Topologically nonplanar flows must exit a catenoid.
The paper proves convexity of certain solitons and expanders in high dimensions.
problem Proving convexity of specific solitons and expanders in Rn+1. method Inspired by Spruck-Xiao and Derdziński, the paper uses geometric analysis to prove convexity.
result The paper proves the convexity of complete 2-convex translating and expanding solitons and expanders in Rn+1 for n≥3. Convexity preserved in curved surfaces moving at concave speeds.
problem Deforming convex surfaces with concave speeds.
method Nonlinear geometric flows with concave speed functions.
result High curvature regions remain approximately convex.
The paper studies a curve flow preserving anisotropic length for convex curves, leading to a homothetic limit.
problem Anisotropic length preservation in curve deformation.
method A curve flow that maintains anisotropic length, analyzed for convex closed curves.
result Convex curves evolve to homothetic limits of Wulff shapes as time approaches infinity.
Study flow on de Sitter space for convex hypersurfaces.
problem Behavior of locally constrained inverse curvature flow in de Sitter space.
method Analyze flow on de Sitter space with specific initial conditions and inequalities.
result Derive Alexandrov-Fenchel type inequalities.
We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier S. Here S can be any properly embedded, oriented surface in Rn+1 of bounded geometry. We also give an alternative proof that convex mean curvature flows with …
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
The paper examines flows that preserve area and length in hyperbolic geometry.
problem Preserving area and length in hyperbolic geometry.
method Inverse curvature flows for convex curves in hyperbolic plane.
result The flows converge to geodesic circles under certain conditions.
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. Study shows thresholding scheme converges for mean curvature flow of convex sets.
problem Analyzing convergence of thresholding scheme for mean curvature flow.
method Time discretization using Merriman, Bence and Osher's scheme, focusing on two-phase mean convex settings.
result Time-integrated energy of approximation converges to limit's energy in minimizing movements interpretation.
Paper relaxes convexity assumptions in mean curvature flow results.
problem Relaxing convexity assumptions in mean curvature flow results.
method Proves a generalized Harnack inequality and uses maximum principle.
result Characterizes family of shrinking spheres for ancient solutions.
Gradient flow expands curves to round shapes.
problem Expanding curves to round shapes.
method Steepest descent L2-gradient flow of entropy.
result Flow converges to a round expanding circle for various initial curves.
A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in R2n carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satis…