Proves optimal isoperimetric inequality in de Sitter space.
problem Optimal isoperimetric inequality for specific hypersurfaces in de Sitter space.
method Analyzes spacelike, compact, star-shaped, and 2-convex hypersurfaces in de Sitter space.
result Proves an optimal isoperimetric inequality for the specified hypersurfaces.
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.
The paper discusses mean curvature flow with surgeries for 2-convex hypersurfaces, focusing on neck detection and gluing.
problem Mean curvature flow with surgeries for 2-convex hypersurfaces.
method Establishing neck detection, gluing cross sections, and using harmonic spherical parametrisation.
result Uniqueness, existence, and overlapping properties for normal parametrisations on (ε,k)-cylindrical hypersurface necks. In [7], Guan, Ren and Wang obtained a C2 a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation σ2(κ(X))=f(X,ν(X)). In this note, we give a simpler proof of this result, and extend it to space forms.
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.…
The paper solves a conjecture about spacelike hypersurfaces in de Sitter space.
problem Proving an Alexandrov-Fenchel inequality for closed 2-convex spacelike hypersurfaces in de Sitter space.
method Investigating the locally constrained inverse curvature flow to establish the inequality.
result Established an Alexandrov-Fenchel inequality for closed 2-convex spacelike hypersurfaces in de Sitter space.
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 new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in RN, as announced in arXiv:1304.0926. Our proof works for all N≥3, including mean convex surfaces in R3. We also derive a priori estimates for a more general class of flows in a local and flexible setting.
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.
The article extends mean curvature flow with surgery for low entropy hypersurfaces.
problem Extending mean curvature flow with surgery for hypersurfaces with low entropy.
method Mean curvature flow with surgery for mean convex hypersurfaces with entropy less than Λn−2, without assuming 2-convexity. result Smooth n-dimensional closed self shrinkers with entropy less than Λn−2 are isotopic to the round n-sphere. 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. The study proves non-orientable surfaces can map to a torus.
problem Embedding non-orientable surfaces in 4D space.
method Proving mapping to a 2D torus for 2-convex surfaces.
result Projective plane and Klein bottle cannot be 2-convex in 4D space.
Proves path-connectedness of 2-convex embedded spheres moduli space.
problem Path-connectedness of moduli space of 2-convex embedded spheres.
method Mean curvature flow with surgery.
result Proves path-connectedness for every n.
We prove that any translating soliton for the mean curvature flow which is noncollapsed and uniformly 2-convex must be the rotationally symmetric bowl soliton. In particular, this proves a conjecture of White and Wang, in the 2-convex case in arbitrary dimension.
Penrose conjecture proven for specific initial data sets.
problem Proving Penrose conjecture for certain types of initial data sets.
method Used σ-inverse mean curvature flow and a monotonicity formula.
result Penrose conjecture established for 2-convex initial data sets.
Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.
problem Understanding the nature and behavior of singularities in level set flow.
method Analytical approach using Lojasiewicz inequality and curvature blow-up rates.
result The arrival time is C2 near a critical point if and only if it satisfies a Lojasiewicz inequality. New method for high-dimensional submanifolds using surgery and curvature control.
problem Mean curvature flow in high codimension with topological control.
method Mean curvature flow with surgery, new a priori estimates for second fundamental form.
result Sharp classification of quadratically 2-convex submanifolds in higher codimensions.
Proves Penrose inequality with charge for 2-convex initial data sets.
problem Proving Penrose inequality with charge for 2-convex initial data sets.
method Uses Dong's 2-convexity condition and P-inverse mean curvature flow, modifying monotonicity formula for charge term.
result Establishes Penrose inequality with charge for 2-convex initial data sets.
Constructs 2-convex functions approximating distances in Alexandrov spaces.
problem Distance approximation in finite-dimensional Alexandrov spaces.
method Constructs 2-convex functions in Alexandrov spaces.
result Functions can be lifted to close Alexandrov spaces.
Paper studies minimal hypersurfaces and their impact on compact manifolds with nonnegative scalar curvature.
problem Analyzing the boundary behavior of compact manifolds with nonnegative scalar curvature.
method Examines the effect of minimal hypersurfaces on the boundary of compact manifolds.
result Establishes an inequality relating mass, area of minimal hypersurfaces, and weighted total mean curvatures.
The paper connects convex functions to p-subharmonic functions and proves their equivalence.
problem Understanding the relationship between convex functions and p-subharmonic functions.
method Average principle, variational methods, and PDE techniques.
result Convex functions on R^n are p-subharmonic for every p > 1.
Proves certain subgroups of genus 2 handlebody group are convex cocompact.
problem Characterizing subgroups of genus 2 handlebody group.
method Proving convex cocompactness of purely pseudo-Anosov subgroups.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact.
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
The paper strengthens a singularity theorem in General Relativity.
problem Proving conditions under which spacetime is incomplete or has specific geometric structures.
method Improving a previous theorem by Galloway and Ling, the paper introduces new conditions for spacetime properties.
result Conditions for spacetime to be past null geodesically incomplete, or have specific geometric structures.
An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to $\bR^3$. Hence, M is irreducib…
An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformati…
The article uses surgery on mean curvature flow to study level set flow's regularity and stability.
problem Analyzing the regularity and stability of level set flow.
method Using mean curvature flow with surgery to derive estimates.
result Demonstrates stability of the plane under level set flow.
The study finds billiard trajectories with infinitely many reflections in certain cones.
problem Existence of billiard trajectories with infinitely many reflections.
method Analysis of C3 convex cones and elliptic cones in R3. result Existence of C2 convex cones with billiard trajectories having infinitely many reflections. A (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R3 with transition maps in the affine transformation group Aff(R3). We will show that a connected closed affine 3-manifold is either an affine Hopf 3-manifold or decomposes canonically to conca…
Smoothly bounded domains have special functions that are plurisubharmonic.
problem Finding smooth functions that are plurisubharmonic on bounded domains.
method Proving existence of smooth defining functions that are p-plurisubharmonic. result Smooth domains with smooth p-convex boundaries admit smooth defining functions that are p-plurisubharmonic. An (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R3 with transition maps in the affine transformation group Aff(R3). Equivalently an affine 3-manifold is a 3-manifold with a flat torsion-free affine connection. We show that a closed affine 3-mani…
Two-stage nonconvex algorithm and convex relaxation both achieve optimal accuracy in noisy blind deconvolution.
problem Solving bilinear systems of equations with random noise under different designs.
method Two-stage nonconvex algorithm and convex relaxation.
result Both methods achieve minimax-optimal accuracy in the presence of random noise.
We supply a proof of the fact that a hyperbolic 3-manifold M with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion Mi of M and modify the boundary to make them 2-convex. We use the induced path-metric, wh…
The paper proves a Feynman-Kac formula for differential forms on manifolds with boundary.
problem Constructing harmonic forms from bounded ones on manifolds with positivity properties.
method Using a Feynman-Kac formula for differential forms on manifolds with boundary.
result Geometric obstruction to the existence of metrics with 2-convex boundary and positive R2. Using PL-methods, we prove the Marden's conjecture that a hyperbolic 3-manifold M with finitely generated fundamental group and with no parabolics are topologically tame. Our approach is to form an exhaustion Mi of M and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the s…
The paper characterizes hypersurfaces in curved spaces using their geometry.
problem Geometric characterization of hypersurfaces in curved spaces.
method Extrinsic geometry analysis of conformally and radially flat hypersurfaces.
result Classification of hypersurfaces in terms of rotation and semi-parallel hypersurfaces.
Study on lightlike hypersurfaces in metallic semi-Riemannian manifolds.
problem Exploring geometric properties of lightlike hypersurfaces in metallic semi-Riemannian manifolds.
method Investigation of invariant and screen semi-invariant lightlike hypersurfaces, examination of integrability conditions.
result Induced structure on invariant lightlike hypersurfaces is metallic.
The paper classifies Codazzi hypersurfaces and characterizes minimal hypersurfaces in Nil^4.
problem Classifying hypersurfaces in Nil^4.
method Using Lie group structure and Codazzi conditions.
result Characterization and classification of minimal hypersurfaces in Nil^4.
Classification of hypersurfaces in homogeneous spaces with specific properties.
problem Classifying hypersurfaces in Riemannian homogeneous spaces with additional assumptions.
method Analyzing hypersurfaces under various conditions in homogeneous spaces CP3. result All extrinsically homogeneous hypersurfaces are classified in all homogeneous CP3 spaces. Study on Hopf hypersurfaces in complex Grassmannians, proving constant Reeb curvature.
problem Proving properties of Hopf hypersurfaces in complex Grassmannians.
method Analyzing real hypersurfaces, proving nonexistence of certain foliations, and classifying hypersurfaces.
result Constant Reeb curvature for Hopf hypersurfaces in complex Grassmannians of rank two.
Study Hopf hypersurfaces in geodesic spaces, proving conditions for tangential convex hypersurfaces to be Hopf.
problem Characterizing Hopf hypersurfaces in geodesic spaces.
method Analyzing Hopf hypersurfaces in (para-)Kaehler manifolds and canonical structures of geodesic spaces.
result Tangential convex hypersurfaces are Hopf in geodesic spaces with respect to canonical structures, except in 3D where a second structure applies.
Study of spacelike Dupin hypersurfaces in Lorentzian space forms.
problem Characterizing and classifying spacelike Dupin hypersurfaces in Lorentzian space forms.
method Using conformal geometry, the study classifies hypersurfaces with constant Möbius curvatures.
result Classification of spacelike Dupin hypersurfaces with constant Möbius curvatures.
The paper studies special null hypersurfaces in spacetimes.
problem Characterizing null screen isoparametric hypersurfaces in Lorentzian space forms.
method Developed screen isoparametric hypersurface concept for null hypersurfaces of Robertson-Walker spacetimes, derived Cartan identities, and provided local characterizations.
result Derived Cartan identities for the screen principal curvatures of null screen hypersurfaces in Lorentzian space forms and provided a local characterization.
Study on biconservative hypersurfaces with constant scalar curvature in space forms.
problem Characterize biconservative hypersurfaces with constant scalar curvature in space forms.
method Analyzing biconservative hypersurfaces in space forms Nn+1(c), proving properties and finding specific examples. result Proves that biconservative hypersurfaces with constant scalar curvature in N4(c) have constant mean curvature, and in N5(c), they are either rotational or constant mean curvature. The paper identifies unique minimal hypersurfaces in space forms.
problem Finding minimal hypersurfaces in space forms.
method Analyzing Simons' equation to identify minimal hypersurfaces.
result Catenoids and Clifford minimal hypersurfaces are the only complete minimal hypersurfaces satisfying Simons' equation in space forms.
The paper classifies various types of hypersurfaces in a product space.
problem Classifying hypersurfaces in a specific product space.
method Analyzing hypersurfaces with constant curvatures, product angle functions, and additional conditions.
result Different types of hypersurfaces are classified based on their properties.
The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.
problem Characterizing and understanding Laguerre isotropic hypersurfaces.
method Analyzing hypersurfaces with zero Laguerre form and constant eigenvalues of the Laguerre tensor.
result For L-isotropic hypersurfaces, if they are also L-isoparametric, the constant λ must be zero. Classifies hypersurfaces with constant isotropic curvature in space forms.
problem Identifying hypersurfaces with constant isotropic curvature in space forms.
method Analyzing complete orientable hypersurfaces and their properties.
result Hypersurfaces have constant mean curvature only if they are isoparametric, and are minimal under specific conditions.