The paper studies Legendrian mean curvature flow in η-Einstein Sasakian manifolds.
problem Existence and asymptotic behavior of Legendrian curves in η-Einstein Sasakian manifolds.
method Legendrian mean curvature flow, stability condition, Thomas-Yau conjecture.
result Existence and asymptotic convergence of long-time solutions.
The article recovers the Smale conjecture on a Sasakian 3-sphere using Legendrian mean curvature flow.
problem Recovering the Smale conjecture on a Sasakian 3-sphere.
method Using Legendrian mean curvature flow to deform area-preserving contactomorphisms to isometries.
result Obtained the minimal Legendrian graph in S² × S³.
The paper introduces a new flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.
problem Investigating the behavior of Legendrian curves in specific geometric settings.
method Introducing and analyzing a modified inverse mean curvature flow for Legendrian curves in Sasakian sub-Riemannian 3-manifolds.
result The flow preserves the Legendrian condition and increases the length of curves, with specific asymptotic behaviors.
We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus o…
We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve of the 3-sphere or a Legendrian curve of the anti de Sitter 3-sp…
Motivated by Legendrian curve shortening flows in R3, we study the curve shortening flow of figure-eight curves in the plane. We show that, under some symmetry and curvature conditions, a figure-eight curve will shrink to a point at the first singular time.
Study geometric mKdV flows for Legendrian curves in a 3-sphere.
problem Investigate geometric evolution equations for Legendrian curves.
method Define a symplectic structure and show mKdV and associated flows.
result Show mKdV equation as curvature evolution induced by Hamiltonian flows.
The paper extends sphere theorems to higher-order mean curvature functions on specific hypersurfaces.
problem Proving sphere theorems for hypersurfaces with W2,n regularity. method Extending Montiel-Ros argument and using Legendrian cycles.
result Proves existence of n-dimensional Legendrian cycles with 2n-dimensional support. The study constructs a Legendrian cycle for FnW2,n-sets and proves Reilly-type variational formulae.
problem Understanding higher-order mean curvature integrals of non-smooth sets.
method Construction of a Legendrian cycle and analysis of proximal unit normal bundles.
result Reilly-type variational formulae for higher-order mean curvature integrals of FnW2,n-sets. Legendrian arcs connect veering triangulations to Anosov flows.
problem Connecting veering triangulations to Anosov flows for study.
method Realizing edges as Legendrian arcs with a bicontact structure.
result Veering triangulations can be placed in steady position.
We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using …
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow an…
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
Generalizes surgery techniques for projectively Anosov flows.
problem Creating new projectively Anosov flows from existing ones.
method Introducing a generalized Goodman surgery technique for projectively Anosov flows.
result Generates new examples of projectively Anosov flows on hyperbolic 3-manifolds.
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.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
Alexandrov immersed surfaces maintain their properties under mean curvature flow.
problem Preserving Alexandrov immersivity in mean curvature flow.
method Mean curvature flow techniques adapted for Alexandrov immersed, 2D surfaces.
result Mean curvature flow properties hold for Alexandrov immersed surfaces.
The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.
problem Analyzing mean curvature flow in a Ricci flow background.
method Computing variational properties and deriving evolution equations for mean curvature and second fundamental form.
result Established a Huisken's monotonicity-type formula for mean curvature solitons in an extended Ricci flow.
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
Ancient solutions to mean curvature flow have unique shapes.
problem Understanding unique shapes of ancient solutions.
method Proved a Bernstein theorem for ancient solutions.
result Ancient solutions to mean curvature flow have unique shapes.
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we will prove the existence and uniqueness for solut…
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.
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 paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended thi…
Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
Proves existence and uniqueness of mean curvature flow.
problem Existence and uniqueness of mean curvature flow.
method Heat kernel estimates and contraction mapping principle.
result Continuous dependence of mean curvature flow on initial data.
Quantitative estimate for curvature in mean curvature flow.
problem Estimating curvature in mean curvature flow.
method Proving a curvature estimate for smooth convex ancient flows.
result Curvature grows at most quadratically in terms of rescaled extrinsic distance.
Proves existence of shrinkers via mean curvature flow.
problem Existence of shrinkers under mean curvature flow.
method Producing compact, smoothly embedded surfaces that develop singularities under mean curvature flow.
result Proves existence of many shrinkers.
Smooth convergence to equidistant surfaces in Fuchsian manifolds.
problem Smooth convergence of modified mean curvature flow in Fuchsian manifolds.
method Modified mean curvature flow starting from an arbitrary graph in a Fuchsian manifold.
result Equidistant surfaces of constant mean curvature are reached as to∞. We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow wit…
Paper proves Hamilton's pinching theorem using mean curvature flow.
problem Hamilton's pinching theorem in extrinsic geometry.
method Mean curvature flow approach.
result Proof of Hamilton's pinching theorem.
Proves Hamilton's theorem using mean curvature flow.
problem Compactness of pinched hypersurfaces with bounded curvature.
method Mean curvature flow to prove Hamilton's theorem.
result Rigorous proof of Hamilton's theorem.
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.
No breathers found for mean curvature flow in noncompact space.
problem Existence of breathers in mean curvature flow.
method Proved no breather theorems for mean curvature flow in noncompact Euclidean space.
result Breathers must be solitonic solutions under certain conditions.
Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.
problem Mean curvature flow of totally real submanifolds.
method Quantitative analysis of almost minimal submanifolds.
result Established convergence result for mean curvature flow.
New study confirms some mean curvature flow solutions have bounded mean curvature.
problem Existence of mean curvature flow singularities with bounded mean curvature.
method Construction of specific solutions in RN for N≥8. result A nontrivial subset of solutions has uniformly bounded mean curvature.
Ancient symplectic solutions to mean curvature flow are flat.
problem Understanding ancient solutions to mean curvature flow in symplectic geometry.
method Using a complex phase map to prove a Bernstein theorem.
result Ancient solutions to the symplectic mean curvature flow are flat.
Proves uniqueness of blowups for forced mean curvature flow.
problem Proving uniqueness of blowups for forced mean curvature flow.
method Adapting methods from Euclidean space mean curvature flow to handle forcing term and blow-up limits.
result Uniqueness of tangent cones for forced mean curvature flow at self-shrinkers and cylindrical self-shrinkers.
For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementary proof of Small's formula. Moreover, a similar formula for Legendrian curves in PSL(2,C) is given. As null curves in PSL(2,C) are related …
Ancient solutions found for mean curvature flows in hyperbolic spaces.
problem Mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
method Study mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
result Ancient solutions always exist for these flows.
Study proves existence and uniqueness of ancient flows from cones.
problem Existence and uniqueness of ancient rescaled mean curvature flows.
method Proved existence and uniqueness using strong uniqueness theorem.
result Proved existence and uniqueness of ancient flows from cones.
Paper proves Harnack inequality for f-mean curvature flow.
problem Proving Harnack inequality for f-mean curvature flow. method Gradient flow of the weighed area functional with measure density function e−f. result Proves Li-Yau-Hamilton type Harnack estimate.
Ancient flows converge fast with finite curvature and convexity.
problem Understanding ancient mean curvature flows with finite curvature.
method Established exponentially fast convergence and finite curvature properties.
result Ancient flows have finite total curvature and finite mass drop.
In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqu…
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.
Constructs approximate mean curvature flows for general varifolds.
problem Mean curvature flow for general initial data.
method Approximation of mean curvature flows using varifolds and iterated push-forwards.
result Approximate mean curvature flow converges to a spacetime Brakke flow under certain conditions.
Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.
problem Rigidity of ancient solutions to mean curvature flow with convex Gauss image.
method Refined curvature estimates.
result Better rigidity theorems for ancient solutions in higher codimension.