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. Unbounded convex domains have zero mean curvature on disconnected boundaries.
problem Understanding mean curvature in unbounded convex domains.
method Analyzing mean curvature on disconnected boundary components.
result Mean curvature is zero on disconnected boundary components of unbounded mean convex domains.
Strict convexity of graphs with constant mean curvature is proven under certain conditions.
problem Proving strict convexity of graphs with constant mean curvature.
method Analyzing the Dirichlet problem for graphs with normalized constant mean curvature and planar boundary.
result The optimal solvability condition for the mean curvature of the boundary suffices to prove the strict convexity of the graph.
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.
Smooths out complex shapes into simpler forms.
problem Transforming complex shapes into simpler, smooth forms.
method Perturbing minimizing hypercones and viscosity mean convex cones into smooth, properly embedded hypersurfaces.
result Properly embedded smooth minimizing hypersurfaces and self-expanders are achieved.
The study proves properties of translating solitons in 3D space.
problem Characterizing translating solitons in R3. method Analyzing mean curvature flow and mean convex translating solitons.
result Proves convexity of complete immersed translating solitons and classifies them.
Proves convexity of certain hypersurfaces with negative λ.
problem Understanding convexity of hypersurfaces with specific λ values.
method Analyzes mean convex hypersurfaces and proves convexity for λ ≤ 0.
result Closed n-dimensional mean convex λ-hypersurfaces are convex if λ≤0. Flow doesn't get wider near singularities if they're convex.
problem Preventing the fattening of surfaces during flow.
method Analyzing mean curvature flow with mean convex singularities.
result The level set flow of a mean convex initial surface doesn't get wider near singularities.
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. 2-convex translating solitons are locally strictly convex.
problem Characterizing the convexity of translating solitons in mean curvature flow.
method Analyzing uniformly 2-convex translating solitons in Rn+1. result Locally strictly convex translating solitons are axisymmetric.
Study confirms mean convexity of singularity neighborhood in mean curvature flow.
problem Understanding the structure of singularities formed by mean curvature flow.
method Detailed analysis of a small neighborhood around the blowup point.
result The neighborhood is mean convex, confirming a conjecture.
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.
The paper studies how convex surfaces shrink under mean curvature flow with a free boundary.
problem Mean curvature flow of convex surfaces with a free boundary on convex barriers.
method Introduced a new perturbation argument to establish convexity and pinching estimates.
result The flow contracts a sufficiently convex surface to a point in finite time, asymptotic to a half-sphere.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Study shows topological constraints on manifolds with non-negative scalar curvature and mean convex boundary.
problem Topological constraints on manifolds with non-negative scalar curvature and mean convex boundary.
method Constructing examples of compact manifolds that do not admit such metrics.
result Many compact manifolds with boundary do not admit a metric of non-negative scalar curvature and mean convex boundary.
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 …
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.
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. Formula extends mean curvature to convex sets.
problem Extending mean curvature to convex sets.
method Derived a special formula for ovaloids.
result Proved integral formula for ovaloids.
A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if we are given a mean-convex set and a null-homotopic Jordan curve on its boundary, then there exists an embedded minimal disk with boundary the given curve contained in the starting mean-convex set. Does a mean-convex set…
We consider mean-convex Alexandrov embedded surfaces in the round unit 3-sphere, and show under which conditions it is possible to continuously deform these preserving mean-convex Alexandrov embeddedness.
The paper proves inequalities for star-shaped and F-mean convex hypersurfaces in Rn+1.
problem Proving geometric inequalities for specific types of hypersurfaces.
method Using anisotropic p-momentum, perimeter, and volume, the paper derives inequalities for star-shaped and F-mean convex hypersurfaces. result The Wulff shape of F is the unique minimizer of the corresponding functionals among all star-shaped and F-mean convex sets. The study explores conjectures on mean convex domains in spaces with scalar curvature constraints.
problem Understanding mean convex domains in spaces with scalar curvature constraints.
method Formulated conjectures and proved theorems.
result Motivated conjectures on mean convex domains.
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
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.
The paper proves unique ancient solutions to mean curvature flow in higher dimensions are symmetric.
problem Proving uniqueness of ancient solutions to mean curvature flow in higher dimensions.
method Analyzing strictly convex, uniformly two-convex, and noncollapsed ancient solutions.
result Ancient solutions are rotationally symmetric translating solitons.
The article uses harmonic mean curvature flow to prove new geometric inequalities for convex hypersurfaces.
problem Proving new geometric inequalities for convex hypersurfaces in hyperbolic space.
method Harmonic mean curvature flow, Alexandrov-Fenchel inequalities, inverse mean curvature flow, Heintze-Karcher type inequality.
result New geometric inequalities for convex hypersurfaces in hyperbolic space.
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.
The article proves finiteness results for 2D convex hypersurfaces using surgery on mean curvature flow.
problem Proving finiteness for 2D convex hypersurfaces in Rn+1. method Using mean curvature flow with surgery for 2 convex hypersurfaces.
result Proves extrinsic finiteness results in the spirit of Cheeger's compactness theorem.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
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.
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.
We use Ilmanen's elliptic regularization to prove that for an initially smooth mean convex hypersurface in Euclidean n-space moving by mean curvature flow, the surface is very nearly convex in a spacetime neighborhood of every singularity. Previously this was known only (i) for n < 7, and (ii) for arbitrary n up to the…
New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.
problem Finding a Sobolev inequality for mean convex spacelike submanifolds in Minkowski space.
method Applying the ABP estimate method to spacelike submanifolds in Rn,1. result Obtained a Sobolev inequality without a mean curvature term for mean convex hypersurfaces.
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.
New control on diameter and curvature for evolving surfaces.
problem Controlling the diameter and curvature of evolving surfaces under mean curvature flow.
method Detailed analysis of cylindrical regions under mean curvature flow.
result Intrinsic diameter stays uniformly controlled as surfaces approach first singular time.
Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow.…
The study proves splitting theorems for manifolds with specific curvature and boundary conditions.
problem Proving splitting theorems for manifolds with specific curvature and boundary conditions.
method Warped product splitting theorem in manifolds with Ricci curvature bounded from below, requiring parabolic and convex boundary.
result Established splitting results for various manifolds with specific curvature and boundary conditions.
The paper identifies the unique ancient convex solution to mean curvature flow in 3D.
problem Classifying ancient solutions to mean curvature flow in 3D.
method Analyzing rotationally symmetric bowl solitons.
result The rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in 3D that is strictly convex and noncollapsed.
The paper studies how convex hypersurfaces evolve under a specific speed function of the mean curvature.
problem Evolution of convex hypersurfaces under a non-homogeneous speed function.
method Evolution of a closed convex hypersurface in Rn+1 with a speed function depending only on the mean curvature. result The flow exists on a finite maximal interval, convexity is preserved, and the hypersurfaces shrink to a point.
We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…
The study establishes curvature estimates and convexity for a specific type of minimal surfaces.
problem Curvature estimates and convexity for a particular class of minimal surfaces.
method Compactness argument and curvature estimates for a family of surfaces.
result Characterization of convexity for properly embedded minimal surfaces with specific curvature conditions.
Study self-expanding solutions of mean curvature flow in various dimensions.
problem Characterize complete mean convex self-expanding hypersurfaces and their properties.
method Analyzing the function ∣A∣2/∣H∣2 and ∣Aξ∣2/∣H∣2 to understand the structure of self-expanders. result Complete mean convex self-expanders are products of self-expanding curves and flat subspaces under certain conditions.
In this note we announce results on the mean curvature flow of mean convex sets in 3-dimensions. Loosely speaking, our results justify the naive picture of mean curvature flow where the only singularities are neck pinches, and components which collapse to asymptotically round spheres.
We introduce the moduli space of spectral curves of constant mean curvature (\cmc\hspace{-5pt}) cylinders of finite type in the round unit 3-sphere. The subset of spectral curves of mean-convex Alexandrov embedded cylinders is explicitly determined using a combination of integrable systems and geometric analysis techni…
Characterizes Euclidean balls with lower bounded k-th mean curvature.
problem Characterizing convex bodies with lower bounded k-th mean curvature.
method New isoperimetric-type inequality and sharp characterizations.
result Euclidean balls uniquely satisfy the condition.