Study high codimension mean curvature flow in Riemannian manifolds, proving limiting flow in Euclidean space.
problem Analyzing mean curvature flow in high codimension Riemannian manifolds.
method Establishing codimension estimate, using quadratic pinching condition, gradient estimates.
result Existence of limiting flow in Euclidean space under cylindrical pinching condition.
We study high codimension mean curvature flow of a submanifold Mn of dimension n in Euclidean space Rn+k subject to the quadratic curvature condition ∣A∣2≤cn∣H∣2,cn=min{3n4,n−21}. This condition extends the notion of two-convexity for hypersurface…
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.
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.
Study mean curvature flow of high codimension submanifolds in complex projective space.
problem Analyse mean curvature flow of high codimension submanifolds in complex projective space.
method Establish codimension estimate, prove convergence to smooth limiting flow, and prove decay estimate.
result Prove existence of limiting flow under cylindrical type pinching.
The paper improves the approximation of isometric immersions in high codimension.
problem Isometric immersions in high codimension.
method Uniform approximation of short immersions by C1,θ isometric immersions. result Achieved C1,θ regularity for isometric immersions in local settings. Study curve shortening flow in high dimensions with boundary constraints.
problem Understanding the behavior of curves in high-dimensional spaces with boundary conditions.
method Used curvature and higher-derivative estimates, Stahl-type maximum principle, and blow-up analysis.
result Flow converges to a shrinking semicircle model or has only semicircle boundary singularities in low entropy regimes.
We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic behavior of these solutions. Add on remark: the construction in this paper has …
Researchers found counterexamples to a 2-jet determination theorem in higher codimension.
problem Counterexample construction to the 2-jet determination Chern-Moser Theorem in higher codimension.
method Constructed counterexamples of quadratic submanifolds with specific properties.
result Generated counterexamples to the 2-jet determination Chern-Moser Theorem in higher codimension.
Characterizes projective submanifolds in high dimensions.
problem Characterizing projective submanifolds of specific codimensions.
method Uses Chern classes and ample line bundles.
result Generalizes earlier findings for hypersurfaces.
The purpose of this article is to examine the possible shapes of type I singularities that form in the mean curvature flow of submanifolds of arbitrary codimension, assuming that the initial submanifold satisfies a particular curvature pinching condition.
Study shows instability of specific cone solutions in high-dimensional spaces.
problem Unstable solutions of minimal graphs in high codimension.
method Min-max technique applied to Euclidean spaces.
result First examples of non-smooth unstable minimal graphs.
We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown …
The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study …
This is a very brief report on recent developments on the Dirichlet problem for the minimal surface system and minimal cones in Euclidean spaces. We shall mainly focus on two directions: (1) Further systematic developments after Lawson-Osserman's paper \cite{l-o} on the Dirichlet problem for minimal graphs of high codi…
The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.
problem Mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.
method Analysis of natural curvature pinching and noncollapsing quantities under mean curvature flow.
result The mean curvature flow deforms any initial spacelike-convex submanifold to a point in finite time, and is asymptotic to a shrinking sphere in a maximally spacelike subspace.
The paper extends a theorem about stable minimal surfaces to higher codimensions.
problem Stability and holomorphicity of parabolic stable minimal surfaces in higher-dimensional spaces.
method Generalization of a classical theorem to higher codimensions, with additional assumptions on the normal bundle.
result Holomorphicity of stable minimal surfaces in higher-dimensional spaces.
Curve shortening flow converges to a point with entropy bound.
problem Analyzing the behavior of curves under shortening flow near singularities.
method Analyzes blow-up limits and uses entropy bounds to prove convergence.
result Initial curves with entropy bound converge to a round point in finite time.
Graphs with bounded anisotropic mean curvature are regular almost everywhere.
problem Understanding the regularity of graphs with anisotropic mean curvature.
method Proving regularity for m-dimensional Lipschitz graphs with anisotropic mean curvature bounded in Lp. result Graphs with bounded anisotropic mean curvature are regular almost everywhere.
The paper characterizes stable cohomotopy groups in codimensions two and three, linking algebraic and geometric perspectives.
problem Characterizing stable cohomotopy groups in specific codimensions.
method Algebraic and geometric approaches, including CW complexes and bordism theory.
result Complete characterizations of stable cohomotopy in codimension two and partial results in codimension three.
In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem characterizi…
Let f: P-->W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram C<--dT-->P. This homotopy push-out square is an example of what is called a Poincare embedding. We study…
We construct Anosov diffeomorphisms on manifolds that are homeomorphic to infranilmanifolds yet have exotic smooth structures. These manifolds are obtained from standard infranilmanifolds by connected summing with certain exotic spheres. Our construction produces Anosov diffeomorphisms of high codimension on infranilma…
Simple criteria for codimension two surface singularities.
problem Identifying singularities in surfaces of codimension two.
method Provided criteria for singularities in surfaces of codimension less than or equal to two.
result Conditions for codimension two singularities in ruled surfaces and center maps.
Generalizes halfspace theorems to higher dimensions for self-shrinkers.
problem Limitations of halfspace theorems in higher dimensions for self-shrinkers.
method Extends codimension 1 results to arbitrary codimension.
result Establishes new halfspace theorems for self-shrinkers in arbitrary codimension.
Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.
problem Discreteness of constant scalar curvatures of compact minimal submanifolds in unit spheres.
method Refined Simons' first gap theorem and Yau's theorems for high-codimensional submanifolds.
result Lu's conjecture for minimal 2-spheres and surfaces proved under inequality conditions.
Alternative approach to rigidity of high-dimensional isometric immersions.
problem Rigidity of high-dimensional isometric immersions between compact manifolds.
method Quantitative rigidity estimates, reducing to Euclidean setting and applying Friesecke-James-Müller rigidity estimate.
result Quantitative results showing close proximity to isometric immersions for small stretching and bending energy.
Totally geodesic submanifolds in hyperbolic space up to codimension two.
problem Characterizing minimal homogeneous submanifolds in hyperbolic spaces.
method Analyzing properties of minimal submanifolds in hyperbolic spaces up to codimension two.
result Minimal homogeneous submanifolds of hyperbolic space up to codimension two are totally geodesic.
Sharp curvature pinching for mean curvature flow in spheres proved.
problem Proving sharp curvature pinching for mean curvature flow in spheres.
method Using blow-up arguments, codimension and cylindrical estimates, and rescaling.
result Smooth convergence to a totally geodesic limit in infinite time.
Study wall singularities in spaces with upper curvature bounds.
problem Understanding singularities in spaces with curvature constraints.
method Geometric structure theorem and geometric characterization for codimension one and two.
result Necessary and sufficient conditions for singular sets to be of codimension at least two.
In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by potential theory, more sophisticated methods have to be applied for codimension…
New examples of non-homeomorphic foliation leaves found.
problem Finding non-homeomorphic foliation leaves in manifolds.
method Examples of 5-manifolds and foliations of 6-manifolds.
result Examples of non-homeomorphic foliation leaves in C1 and C∞ foliations. Uniform waist inequalities proven for manifolds with Kazhdan groups in codimension two.
problem Proving uniform waist inequalities for manifolds with specific group properties.
method Using finite covers and Cheeger inequality for manifolds with Kazhdan fundamental groups.
result Finite covers of manifolds with Kazhdan groups satisfy uniform waist inequalities in codimension two.
Stabilization operation for high-dimensional contact manifolds, proving many links are non-simple.
problem Understanding the structure and properties of high-dimensional contact manifolds.
method Definition and proof of stabilization operation for codimension 2 contact submanifolds in dim≥5 contact manifolds. result Many transverse links are non-simple.
The paper finds non-isotopic Legendrian unit conormal bundles in high dimensions.
problem Identifying non-isotopic Legendrian unit conormal bundles in high dimensions.
method Defined strip Legendrian contact homology and coproduct for Legendrian submanifolds.
result Found non-isotopic Legendrian unit conormal bundles using topological and string topology methods.
Confirming a conjecture, we show fundamental groups of certain abelian differentials are framed mapping class groups.
problem Confirming a 1997 conjecture about fundamental groups of abelian differentials.
method Algebraic-geometric approach using Shimada's techniques for computing fundamental groups via morphisms of varieties.
result Orbifold fundamental groups of these strata are described as framed mapping class groups.
Paper constructs a transfer map for codimension 2 submanifolds in higher index theory.
problem Higher index theory of codimension 2 submanifolds.
method Construction of codimension 2 transfer map and adjoint relationship with cyclic cohomology.
result Established adjoint relationship between codimension 2 transfer map and co-transfer map in cyclic cohomology.
New bounds on manifold widths and essential curves in high dimensions.
problem Bounding the l∞-widths of submanifolds in Euclidean space. method Introducing a new approach to systolic geometry involving non-linear complexes and averaging over isometries.
result Proved upper bounds on l∞-widths and existence of essential curves in cubes. The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse…
We construct and embedding of a Nöbeling space Nn−2n of codimension 2 into a Menger space Mn−2n of codimension 2. This solves an open problem stated by R.~Engelking in 1978 in codimension~2.
This paper classifies embedded, codimension-one spheres which are null homotopic. This information is used to show that all null homotopic, immersed codimension-one spheres which are taut in the sense of Terng and Thorbergsson are actually distance spheres.
Study how pairs of 1D foliations can be deformed into contact structures.
problem Understanding deformations of pairs of 1D foliations.
method Linear deformations of pairs of codimension one foliations into contact pairs.
result Main result provides applications and insights into foliation deformations.
We classify irreducible polar foliations of codimension q on quaternionic projective spaces HPn, for all (n,q)=(7,1). We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on HPn are homogeneous if and only if n+1 is a prime number (resp. n is ev…
In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kahler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states th…
Study of codimension-1 embeddings in 3-manifolds using twist maps and push maps.
problem Embedding 3-manifolds in specific topological spaces.
method Sphere twist maps and push maps to construct codimension-1 spun embeddings.
result Every closed orientable 3-manifold admits a codimension-1 spun embedding in specific spaces.
This paper extends NCFI to odd codimension and computes examples.
problem Extending NCFI to foliations of odd codimension.
method Computing NCFI for various foliated manifolds in both even and odd codimensions.
result NCFI is an invariant of foliations in odd codimension, requiring an odd \(K_1\)-class.
Study shows singular set of certain graphs has codimension 1.
problem Understanding the singular set of specific graph structures.
method Proved using the area stationarity condition.
result Singular set has codimension 1.
In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with ∣A∣2≤1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.