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.
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.
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. 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. 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.
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.
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 …
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. 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.
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.
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…
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. 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.
We prove that any complete immersed two-sided mean convex translating soliton Σ⊂R3 for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in R3 is the axisymmetric "bowl soliton". We also show that if the mean curvature of…
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.
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.
We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singul…
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.
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.
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.
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.
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.
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.…
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 study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex bound…
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…
We prove that any complete immersed globally orientable uniformly 2-convex translating soliton Σ⊂Rn+1 for the mean curvature flow is locally strictly convex. It follows that a uniformly 2-convex entire graphical translating soliton in Rn+1,n≥3 is the axisymmetric "bowl soliton…
New compact mean convex hypersurfaces found for positive λ.
problem Finding compact embedded hypersurfaces for positive λ.
method Constructing compact mean convex hypersurfaces diffeomorphic to spheres.
result No compact convex embedded λ-hypersurfaces except a round sphere for λ > 0.
We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp Ln−1-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis…
We show that any strictly mean convex translator of dimension n≥3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature…
We consider the mean curvature flow of compact convex surfaces in Euclidean 3-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on the geometry of the barrier, the flow contracts the surface to a point in finite…
For any n-dimensional smooth manifold Σ, we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in Σ are cylindrical (of convex type) if the flow converges to a smooth hypersurface M∞ (maybe empty) at infinity. Previously this was shown (i) for n≤7,…
Convexity preserved in curved surfaces moving at concave speeds.
problem Deforming convex surfaces with concave speeds.
method Nonlinear geometric flows with concave speed functions.
result High curvature regions remain approximately convex.
Study on estimating Gaussian mean from coarse data, resolving identifiability and computational efficiency questions.
problem Estimating the mean of a Gaussian distribution from coarse data (sets containing true samples rather than exact values).
method Analyzes the conditions for mean identifiability and computable estimation under convex partitions.
result Resolves the identifiability and computational efficiency questions for Gaussian mean estimation from coarse data.
Solves a specific Dirichlet problem for Lagrangian mean curvature equations.
problem Solving the Dirichlet problem for Lagrangian mean curvature equations.
method Solves the Dirichlet problem for Lagrangian mean curvature equations on uniformly convex domains.
result Solves the Dirichlet problem for Lagrangian mean curvature equations.
The study examines mean curvature flow and Heegaard surfaces in lens spaces.
problem Analyzing mean convex two-spheres and Heegaard tori in lens spaces.
method Proves path-connectedness of moduli spaces of mean convex two-spheres and Heegaard tori in manifolds with nonnegative Ricci curvature.
result There are always either one or two path components of mean convex Heegaard tori in lens spaces, depending on the ambient manifold's homotopy type.
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumblin…
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
Classifies regularity for Lagrangian mean curvature type equations.
problem Classifying regularity for Lagrangian mean curvature type equations.
method Generalized constant rank theorem for Legendre transform, constructed convex solutions, and showed regularity conditions.
result Optimal regularity conditions for Lagrangian mean curvature type equations.