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…
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48 results for “Mean Curvature Flow”
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.
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 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.
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 for .
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.
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 -mean curvature flow.
problem Proving Harnack inequality for -mean curvature flow.
method Gradient flow of the weighed area functional with measure density function .
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.
Approximate mean curvature flows of a general varifold, and their limit spacetime Brakke flowmath.DG
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.
In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumptio…
Study shows uniform decay rate for singular mean curvature flows.
problem Understanding singularities in mean curvature flows.
method Rescaled flow analysis near compact singularities.
result Uniform decay order bound for the rescaled flow.
New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theorymath.DG
Survey on mean curvature flow with sphere theorems and Yau rigidity theory.
problem Sphere theorems for submanifolds with arbitrary codimension.
method Recent developments on convergence theorems for mean curvature flow.
result Optimal convergence theorem for arbitrary codimension mean curvature flow.
The paper defines a frequency for mean curvature flow and proves its monotonicity.
problem Backwards uniqueness for solutions of mean curvature flow.
method Defining and proving monotonicity of a parabolic frequency for mean curvature flows.
result Frequency monotonicity implies backwards uniqueness for mean curvature flow solutions.
Symmetry groups help define solitons in curved spaces.
problem Understanding solitons in curved spaces.
method Defined generalized solitons using symmetry groups.
result Affine solutions are self-similar.
Study on surfaces pinched by curvature in space forms converging under specific conditions.
problem Investigating convergence of surfaces pinched by curvature in space forms.
method Proving convergence theorems for surfaces pinched by normal curvature in 4-dimensional space forms.
result Generalizes Baker-Nguyen's convergence theorem for surfaces pinched by curvature.
Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.
problem Constructing compact mean curvature flow solutions with bounded mean curvature.
method Following Velázquez, Guo, Sesum, and Stolarski's arguments, constructing solutions in \(\mathbb{R}^n\) with \(n \geq 8\).
result Compact mean curvature flow solutions with bounded mean curvature in \(\mathbb{R}^n\) are constructed.
Study mean curvature flow into evolving manifold with coupled flows.
problem Analyzing mean curvature flow in evolving Riemannian manifolds.
method Coupling Ricci flow and harmonic map heat flow, calculating variations, and using Harnack expressions.
result Obtained a Huisken monotonicity-type formula for mean curvature flow.
Lectures on mean curvature flow and its related equations.
problem Singularity formation, nonuniqueness, and topological change in motion by mean curvature.
method Analyzes motion by mean curvature flow and related equations.
result Exploration of singularity formation, nonuniqueness, and topological change.
The paper studies the consistency of mean curvature flow via volumetric varifolds.
problem Consistency of mean curvature flow.
method Discretization using volumetric varifolds and derivation of Brakke approximate equality.
result Derivation of a Brakke approximate equality involving varifold masses and approximate mean curvatures.
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.
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.
Rotationally symmetric solutions persist after mean curvature flow starts from a double cone.
problem Understanding the symmetry of solutions to mean curvature flow.
method Analyzing solutions coming out of a double cone.
result Rotationally symmetric solutions persist.
A possible evolution of a compact hypersurface in R^n by mean curvature past singularities is defined via the level set flow. In the case that the initial hypersurface has positive mean curvature, we show that the Brakke flow associated to the level set flow is actually a Brakke flow with equality. We obtain as a conse…