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.
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.
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.
The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.
problem Smoothness of mean curvature flow for generic initial data.
method Long-time existence and uniqueness result for ancient mean curvature flows.
result Smooth mean curvature flow until disappearance in a round point for low-entropy hypersurfaces in 4D.
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.
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.
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 shows how to preserve Lagrangian condition in mean curvature flow on Kim-McCann metrics.
problem Preserving Lagrangian condition in mean curvature flow on Kim-McCann metrics.
method Expressed mean curvature flow within generalized mean curvature flow framework.
result Lagrangian condition is preserved along the flow.
Study of mean curvature flows with conical singularities using mathematical techniques.
problem Understanding the dynamics of mean curvature flows near conical singularities.
method Feynman-Kac formula and invariant cone method for noncompact settings.
result Generic initial perturbations avoid conical singularities in mean curvature flows.
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∞. 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…
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.
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.
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.
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.
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…
Study of bubble-sheet singularity in mean curvature flow.
problem Analyzing a specific type of singularity in mean curvature flow.
method Deriving an asymptotic profile for a neighborhood of the singularity.
result An asymptotic profile for a neighborhood of the singularity is derived.
Ancient mean curvature flows get codimension bounds from their tangent flow.
problem Understanding the limiting behavior of ancient mean curvature flows.
method Proving codimension bounds using the tangent flow at −∞. result Ancient mean curvature flows are rigid to their tangent flow at −∞. Theory developed for surfaces moving by mean curvature flow with boundary.
problem Moving surfaces with boundary by mean curvature flow.
method Elliptic regularization for existence, boundary regularity under mild hypotheses.
result General existence theorem and boundary regularity at all positive times.
Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.
problem Entropy for submanifolds in Riemannian manifolds.
method Entropy defined and shown to be monotone along mean curvature flow.
result Partial regularity of mean curvature flow limits of surfaces.
Mean curvature flow shows singularities on smooth surfaces.
problem Understanding singularities in mean curvature flow.
method Analyzing spherical or nondegenerate neck pinches.
result First singular time has isolated singularities.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
This paper studies mean curvature flows near cylindrical singularities.
problem Understanding the behavior of mean curvature flows near cylindrical singularities.
method Proved the rescaled flow converges to a graph over a cylinder, defined nondegeneracy, and showed properties of nondegenerate singularities.
result Nondegenerate cylindrical singularities are isolated, have a mean convex neighborhood, and are type-I.
Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.
problem Longtime existence of mean curvature flow in GRW spacetimes.
method Proved longtime existence using perpendicular Neumann boundary condition and null convergence condition.
result Metric of solution is conformal to GRW leaf's metric in asymptotic time.
Proves multiplicity one for mean curvature flow singularities.
problem Understanding singularities in mean curvature flow of surfaces.
method Analyzes self-shrinkers and constructs perturbations.
result Proves multiplicity one for generic singularities.
Mean curvature flow evolves isometrically immersed base manifolds M in the direction of their mean curvatures in an ambient manifold Mˉ. If the base manifold M is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of ar…
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.
Study finds unique self-expanders for mean curvature flow.
problem Finding solutions to mean curvature flow.
method Derived equation based on generalized Lawson-Osserman cone and modified equilibria theory.
result Existence and uniqueness of self-expanders proved.
In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H^{k} mean curvature flow, and to do a sui…
We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor φ(r). If φ′(r)>0 and φ′′(r)≥0, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of φ′′(r) and …
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 H∈L∞Llocp, the tangent flow is unique when p=∞ and C is a regular cone. In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time T of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …
Sharp estimate for flow in any dimension.
problem Interior gradient estimate for graphical mean curvature flow.
method Proving sharp interior gradient estimate for area decreasing graphical mean curvature flow in arbitrary codimension.
result Generalized result in arbitrary codimension.
The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.
problem Understanding the behavior of intersections in mean curvature flows.
method Proving new intersection dimension monotonicity results for mean curvature flows, Brakke flows, and level set flows.
result The dimension of the intersection of mean curvature flows is non-increasing over time.
Mean curvature flows of hypersurfaces have been extensively studied and there are various different approaches and many beautiful results. However, relatively little is known about mean curvature flows of submanifolds of higher codimensions. This notes starts with some basic materials on submanifold geometry, and then …
In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension d≥1, which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to Le-Šešum \cite{LS} and the authors \cite{XYZ1,XYZ2}. Usin…
In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface Σ is strictly mean convex and star-shaped, then the flow hypersurface Σt converges to a large coordinate sphere as t→∞ exponentially. We also describe an a…
Eigenvalue estimate for shrinkers in mean curvature flow.
problem Eigenvalue estimates on shrinkers for mean curvature flow.
method Generalized earlier work of Ding and Xin to noncompact cases.
result Eigenvalue estimate holds on every properly embedded shrinker.
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. 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…
A family of hypersurfaces evolves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow is the most natural evolution equation in extrinsic geometry, and has been extensively studied ever since the pioneering work of Brakke and Huisken. In the last 15 years, Whi…
In [5], Sáez and Schnürer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature flow. In this paper, we generalize their results to a general curvature setting. Our…
The paper proves new Harnack inequalities for curvature flows in curved spaces.
problem Proving new inequalities for curvature flows in curved spaces.
method Differential Harnack inequalities for flows of strictly convex hypersurfaces by powers of mean curvature in Einstein manifolds.
result New Harnack inequalities for curvature flows in Einstein manifolds with positive sectional curvature.
Generic low-entropy hypersurfaces in 4-6D flow with only generic singularities.
problem Analyzing mean curvature flow of low-entropy hypersurfaces.
method Proving flow encounters only generic singularities for specific entropy conditions.
result Proves flow encounters only generic singularities for low-entropy initial data.
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.
Paper resolves Huisken's conjecture without strict genus drop theorem.
problem Huisken's genericity conjecture in mean curvature flow in R^3.
method Short density-drop theorem + Bamler-Kleiner multiplicity-one theorem for tangent flows.
result Fully resolves Huisken's conjecture without strict genus drop theorem.
Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.
problem Stability of generalized Lagrangian mean curvature flow in cotangent bundle.
method New derivative estimates to weaken initial conditions and remove curvature constraints.
result Stability of flow near special Lagrangian submanifolds in cotangent bundle.
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.