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.
Study of σk flow in Minkowski space for noncompact hypersurfaces.
problem Understanding the behavior of σk curvature flow in Minkowski space. method Proving flow existence and convergence to self-expander after rescaling.
result The flow converges to a self-expander after rescaling.
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.
Study of mean curvature flow on null hypersurfaces leading to MOTS.
problem Detecting marginally outer trapped surfaces (MOTS) in null hypersurfaces.
method Analysis of mean curvature flow on null hypersurfaces with mild conditions.
result Existence and convergence of mean curvature flow to MOTS.
The paper studies a specific centro-affine invariant hypersurface flow in R^(n+1).
problem Existence and uniqueness of a centro-affine invariant hypersurface flow.
method Investigates the flow's existence and uniqueness, explores its properties in centro-affine and Euclidean settings, and investigates long-time behavior.
result The hypersurface converges asymptotically toward an ellipsoid via systematically investigating evolutions of centro-affine invariants.
Minimal hypersurfaces can't always be connected by mean curvature flow.
problem Existence of connecting mean curvature flows for minimal hypersurfaces.
method Minimal hypersurface analogue of gradient flow trajectories between critical points.
result Additional topological and variational obstructions to connecting mean curvature flows.
Ancient mean curvature flows start from unstable minimal hypersurfaces.
problem Constructing ancient solutions to mean curvature flow.
method From an unstable minimal hypersurface with finite total curvature in \(\mathbb{R}^{n+1}\), we construct \(I\)-dimensional families of embedded ancient solutions.
result Ancient solutions arise from unstable minimal hypersurfaces.
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…
The paper studies inverse mean curvature flow on hypersurfaces in space forms.
problem Analyzing the inverse mean curvature flow on hypersurfaces in space forms.
method Investigates the existence and properties of the flow for isoparametric hypersurfaces.
result Characterizes the flow and solutions explicitly under certain conditions.
Study on curvature flow in Minkowski space for cocompact hypersurfaces.
problem Investigating curvature flow in Minkowski space for cocompact hypersurfaces.
method Investigation of the cocompact inverse \(σ_k\) curvature flow in Minkowski space.
result Longtime existence and convergence of the curvature flow established.
We investigate the structure of real hypersurfaces with isometric Reeb flow in Kaehler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.
Study shows how certain hypersurfaces evolve under mean curvature flow.
problem Evolution of isoparametric hypersurfaces under mean curvature flow.
method Reparametrization of the parallel family in short time.
result Evolution given by a reparametrization of the parallel family.
We consider contracting flows in (n+1)-dimensional hyperbolic space and expanding flows in (n+1)-dimensional de Sitter space. When the flow hypersurfaces are strictly convex we relate the contracting hypersurfaces and the expanding hypersurfaces by the Gauss map. The contracting hypersurfaces shrink to a point $x_0…
New flow for capillary surfaces converges to spherical caps.
problem Optimizing capillary surfaces in space forms.
method Constrained mean curvature flow.
result Flow converges to spherical caps globally.
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.
Study anisotropic flow for capillary hypersurfaces, proving new inequalities.
problem Anisotropic capillary hypersurfaces and their properties.
method Anisotropic volume-preserving mean curvature flow, new approach for strictly convex initial hypersurfaces.
result Established new Alexandrov-Fenchel inequalities for strictly convex anisotropic capillary hypersurfaces.
The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.
problem Prove global existence of the Willmore flow in higher dimensions.
method Apply Michael-Simon-Sobolev inequality and Gagliardo-Nirenberg inequalities to establish local energy estimates and maximal existence time.
result Global existence of the Willmore flow in higher dimensions is proven.
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.
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. 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.
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.
The paper studies the stability of volume and area preserving mean curvature flows in Schwarzschild and asymptotic Schwarzschild spaces.
problem Investigating the stability of mean curvature flows in specific spacetime geometries.
method Combining center manifold analysis with global existence results for flows near isoperimetric hypersurfaces.
result Global existence and convergence to constant mean curvature (CMC) hypersurfaces for flows in asymptotic Schwarzschild space.
Study of curvature flow in Minkowski space converging to a hyperboloid.
problem Understanding curvature flows in Minkowski space.
method Analyzing fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space.
result The flow converges to the future timelike hyperboloid after rescaling.
We consider contracting and expanding curvature flows in $\Ss$. When the flow hypersurfaces are strictly convex we establish a relation between the contracting hypersurfaces and the expanding hypersurfaces which is given by the Gauß map. The contracting hypersurfaces shrink to a point x0 while the expanding hypersur…
In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
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.
Avoids noncompact hypersurfaces from touching in evolving flows.
problem Preventing noncompact hypersurfaces from touching in evolving flows.
method Analyzes mean curvature flow and weak set flows in Euclidean and Riemannian spaces.
result Proves that noncompact hypersurfaces remain disjoint in evolving flows.
Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.
problem Preserving positive sectional curvature in contracting curvature flows in hyperbolic space.
method Homogeneous speed flow with positive sectional curvature, including kth mean curvature flow. result Positive sectional curvature is preserved and the hypersurface contracts to a round point in finite time.
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.
Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
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.
The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.
problem Analyzing the behavior of expanding hypersurfaces in Euclidean space.
method Introduced a curvature flow with specific speed function and proved the existence and convergence of the flow under certain conditions.
result The flow converges to a round sphere centered at the origin for all time under specific conditions.
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.
We give a bound on the extinction time for a compact, strictly convex hypersurface in R^{n+1} evolving by a geometric flow where the velocity is given in terms of the curvature. This result generalizes a theorem of Colding and Minicozzi for mean curvature flow solutions to a wider class of flows studied by Ben Andrews.…
Inverse curvature flows shape star-shaped hypersurfaces into spheres.
problem Evolution of star-shaped hypersurfaces inside a convex cone.
method Inverse curvature flows, convexity of the cone, gradient and Hölder estimates.
result Hypersurfaces converge to a round sphere as time goes to infinity.
Study null hypersurfaces in Lorentzian manifolds, proving Riemannian flow structure.
problem Properties of Lorentzian manifolds influenced by totally geodesic null hypersurfaces.
method Coupling rigging technique with null foliation existence to prove Riemann flow structure.
result Proves curvature conditions restrict causal structure of spacetime.
Paper extends isoperimetric inequalities for non-starshaped hypersurfaces.
problem Isoperimetric inequalities for non-starshaped hypersurfaces.
method Volume preserving and area decreasing mean curvature flow with conformal Killing vector fields.
result Established isoperimetric inequalities for a broader class of hypersurfaces.
In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in R3. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in Rn+1 in arbitrary …
Localizes curvature estimates for evolving hypersurfaces under various flows.
problem Establishing curvature estimates for evolving hypersurfaces under different flow conditions.
method Adapted localization of Huisken--Stampacchia iteration method to fully nonlinear flows.
result Asymptotically sharp curvature pinching estimates for general flows.
In this paper we will discuss how one may be able to use mean curvature flow to tackle some of the central problems in topology in 4-dimensions. We will be concerned with smooth closed 4-manifolds that can be smoothly embedded as a hypersurface in R^5. We begin with explaining why all closed smooth homotopy spheres can…
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.
We consider the problem of deforming a one-parameter family of hypersurfaces immersed into closed Riemannian manifolds with positive curvature operator. The hypersurface in this family satisfies mean curvature flow while the ambient metric satisfying the normalized Ricci flow. We prove that if the initial metric of the…
We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over R. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, …
Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.
problem Short time existence and long time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
method Finding sufficient conditions for short time existence and discussing long time existence and convergence.
result Sufficient conditions for short time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space CP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. A…
The paper studies how convex hypersurfaces evolve under curvature flows in space forms.
problem Understanding the evolution of convex hypersurfaces under curvature flows in different space forms.
method Flow by powers of the Gauss curvature in space forms.
result Convex hypersurfaces under the flow by powers of the Gauss curvature in space forms contract to a point in finite time or converge to geodesic spheres.