The paper extends regularity for p-minimizing maps using a Reifenberg Theorem.
problem Quantitative regularity of p-minimizing maps between Riemannian manifolds. method Stratification of singular points based on almost-symmetries, followed by application of a Reifenberg-type Theorem.
result Upper bound on the Minkowski content of the singular set, and k-rectifiability of the singular set. We study generalizations of Reifenberg's Theorem for measures in Rn under assumptions on the Jones' β-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…
Proof of Reifenberg theorem in metric spaces, expanding on Cheeger and Colding's work.
problem Proving the Reifenberg theorem in metric spaces using Gromov-Hausdorff distance.
method Detailed proof of Cheeger and Colding's result, expanding on their arguments.
result BiLipschitz version of the Reifenberg theorem in metric spaces.
The aim of this article is to study effective Reifenberg theorems for measures in a Hilbert or Banach space. For Hilbert spaces, we see all the results from Rn continue to hold with no additional restrictions. For a general Banach spaces we will see that the classical Reifenberg theorem holds, and that a we…
These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another, approximated on all scales by well behaved spaces, typically just Euclidean space its…
The paper proves topological finiteness for surfaces with finite Willmore energy.
problem Understanding the topology of surfaces with finite Willmore energy.
method Combining Allard regularity theorem and Reifenberg's topological disk theorem.
result Topological finiteness for a class of properly immersed surfaces with finite Willmore energy.
This paper improves Reifenberg's theorem for almost calibrated sets, ensuring rectifiability with volume bounds.
problem Improving the rectifiability of sets that are close to subspaces under certain calibrations.
method Using ε-calibrations and positivity conditions, the paper shows that almost calibrated sets are rectifiable with volume bounds.
result Almost calibrated sets are rectifiable with uniform volume bounds.
Unified proof of smooth fibration theorems for collapsed manifolds.
problem Smooth fibration theorems for collapsed manifolds with Ricci curvature bounded below.
method Generalized Reifenberg condition and transformation technique for almost splitting maps.
result Unified proof of smooth fibration theorems in many previous works.
In this paper we study the regularity of stationary and minimizing harmonic maps f:B2(p)⊆M→N between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is kth-stratum of the singular set of f, then it is well known that dimSk≤k, howeve…
In 1960 Reifenberg proved the topological disc property. He showed that a subset of Rn which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-Hölder image of the unit ball in Rm. In this paper we prove that a subset of R3 which is well approximated b…
The paper proves fibration theorems for manifolds with almost nonnegative Ricci curvature.
problem Proving fibration theorems for manifolds with specific curvature conditions.
method Using equivariant regularity theorems and Gromov-Hausdorff convergence.
result Closed manifolds with certain curvature conditions fiber over a b1-torus. We prove existence and regularity of minimizers for Hölder densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams,…
We provide a Reifenberg type characterization for m-dimensional C1-submanifolds of Rn. This characterization is also equivalent to Reifenberg-flatness with vanishing constant combined with suitably converging approximating m-planes. Moreover, a sufficient condition can be given by the finiteness of th…
The study defines a canonical nilpotent structure for certain collapsed manifolds.
problem Understanding the structure of collapsed Riemannian manifolds.
method Analyzes the nilpotent structure of manifolds with bounded Ricci curvature and Reifenberg local covering geometry.
result A canonical nilpotent structure can be defined and uniquely determined over regular limit spaces.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called k-dimensional (ε,R) Reifenberg flat sets in Rn. Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $…
In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in Rn+1 starting from any n-dimensional (ε,R)-Reifenberg flat set with ε sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and …
We study here limit spaces (Mα,gα,pα)→GH(Y,dY,p), where the Mα have a lower Ricci curvature bound and are volume noncollapsed. Such limits Y may be quite singular, however it is known that there is a subset of full measure $\cR(Y)\subseteq Y$, called {\it regular} points, along with c…
Paper provides estimates for varifolds with critical mean curvature.
problem Estimating tilt-excess on varifolds with critical mean curvature.
method Generalizing Lipschitz approximation and Sobolev-Poincaré estimates to almost-integral rectifiable varifolds.
result VMO-type estimates for quadratic tilt-excess on varifolds with critical mean curvature.
The paper proves stability of Ricci flow for certain initial conditions.
problem Stability of Ricci flow for non-smooth initial metrics.
method Analyzes stability of Ricci flows starting from Reifenberg spaces with bounded curvature.
result Smooth three-dimensional, uniformly Ricci-pinched manifolds are either compact or flat.
The paper proves topological stability between RCD spaces and Riemannian manifolds.
problem Proving topological stability between RCD spaces and Riemannian manifolds.
method Using Gromov-Hausdorff distance and regular homeomorphisms, the paper constructs a map between spaces.
result There exists a regular homeomorphism between RCD spaces and Riemannian manifolds under certain conditions.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed …
We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable G chain in ℓ2 is dense in its support, whenever the group G of …
In this paper we first review the covering space method with constrained BV functions for solving the classical Plateau's problem. Next, we carefully analyze some interesting examples of soap films compatible with the covering space method: in particular, the case of a soap film only partially wetting a space curve, a …
Plateau's problem is to find a surface with minimal area spanning a given boundary. In 1960, Reifenberg and Adams developed a definition for "span" using Čech homology, and variants of this definition have been used ever sense. However, limitations of Čech homology resulted in the lack of a natural definition for a bou…
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
This paper studies limits of aspherical manifolds with specific curvature conditions.
problem Understanding the Gromov-Hausdorff limits of aspherical manifolds with given curvature constraints.
method Analyzing sequences of compact manifolds with Ricci curvature or sectional curvature conditions, and using diffeomorphism or homeomorphism properties.
result If the manifolds are diffeomorphic or homeomorphic to nilmanifolds, their limits are also diffeomorphic or homeomorphic to nilmanifolds.
We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and Whit…
Study Ricci flows on manifolds, proving they behave like self-similar solutions and confirming a conjecture.
problem Understanding the behavior of Ricci flows on higher-dimensional manifolds.
method Analyzing n-dimensional Ricci flows with non-negative Ricci curvature, starting at metric cones. result Ricci flows behave like self-similar solutions up to an exponential error in time.
If one considers an integral varifold Im⊆M with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that dimSk≤k. In complete generality nothing else is known about …
A canonical diffeomorphism is constructed for manifolds near spheres.
problem Constructing a canonical diffeomorphism for manifolds near spheres.
method Using the first (n+1)-eigenfunctions of the manifold, a map ildef is constructed and shown to be a diffeomorphism with a uniform bi-Hölder estimate. result The constructed diffeomorphism ildef is canonical and satisfies a uniform bi-Hölder estimate, which is sharp and cannot be improved to a bi-Lipschitz estimate. A mean curvature flow starting from a closed embedded hypersurface in Rn+1 must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded (n−1)-dimensional Lipschitz submanifolds plus a set of dimension at most …
New non-canonical flows found via parabolic Allen-Cahn equations.
problem Existence of non-canonical mean curvature flows inside fattening regions.
method Construction of non-canonical flows as limits of parabolic ε-Allen-Cahn solutions.
result First examples of non-outermost, non-canonical integral Brakke motions.
We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It reveals new, previously unknown, properties that all generalized spaces with a lowe…
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.