Characterizes C1 submanifolds in terms of Reifenberg flatness.
problem Characterizing C1 submanifolds in Rn. method Reifenberg type characterization and integral quotient condition.
result Characterization equivalent to Reifenberg-flatness with approximating planes.
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 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,…
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.
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. $…
Theorem generalizes Reifenberg's for measures with bounds on β-numbers.
problem Bounding measures away from k-rectifiable sets with β-numbers.
method Assumptions on Jones' β-numbers to measure closeness to subspaces.
result Effective measure bounds on μ away from a closed k-rectifiable set.
Introduces methods to approximate complex sets by simpler Euclidean spaces.
problem Approximating complex sets by Euclidean subspaces on all scales.
method Reifenberg theory and Rectifiable Reifenberg theorem.
result Measures can be decomposed into rectifiable and non-rectifiable parts.
The article studies effective Reifenberg theorems in Hilbert and Banach spaces.
problem Understanding measures in Hilbert and Banach spaces.
method Analyzes Reifenberg theorems for measures in Hilbert and Banach spaces, providing conditions for rectifiability.
result Conditions for rectifiability of measures in Banach spaces, including a power gain for uniformly smooth spaces.
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. 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.
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 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. 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 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.
Analyzes soap films using BV functions and covering spaces.
problem Solving Plateau's problem with soap films.
method Covering space method with constrained BV functions.
result Examples of soap films not modelable with Reifenberg method.
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.
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.
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 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 …
Study of collapsed manifolds with bounded Ricci curvature and non-collapsed universal cover.
problem Understanding collapsed manifolds with specific Ricci curvature properties.
method Ricci flow techniques applied to non-collapsed universal cover.
result Partial extension of nilpotent structural results to global Ricci bounded covering geometry.
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.
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.
New epiperimetric inequality for cones with improved regularity results.
problem Regularity of almost area-minimizing currents at singular points.
method Flowing in radial direction for cones with isolated singularities.
result New ε-regularity result for almost area-minimizing currents.
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.
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 …
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.
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.
The paper studies geometric PDEs for flatness on Riemannian manifolds.
problem Understanding flatness in geometric PDEs.
method Study geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness.
result Introduce new Theorems about flatness in Differential Geometry.
Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
problem Characterizing projectively flat metrics on Hopf manifolds.
method Partitioning projectively flat metrics into classes based on Chern scalar curvature sign and proving properties of each class.
result Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.
We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…
The paper characterizes complex Finsler metrics that are projectively flat or dually flat.
problem Characterizing complex Finsler metrics with specific geometric properties.
method Proving conditions for projective flatness and dually flatness in terms of Minkowski metrics.
result Strongly convex complex Finsler metrics are projectively flat or dually flat if and only if they come from Minkowski metrics.
Study flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
New method constructs holonomic immersions from flat submanifolds.
problem Creating holonomic immersions from flat submanifolds.
method Ribaucour transformation and principal coordinate system.
result Holonomic immersions can be constructed using Ribaucour transformation.
Study calculates intersection forms of almost-flat 4-manifolds.
problem Understanding the intersection forms of almost-flat 4-manifolds.
method Calculation of intersection forms for all 4-dimensional almost-flat manifolds.
result Intersection forms of all 4-dimensional almost-flat manifolds have been calculated.
Flat systems of up to 2 dimensions have flat subsystems.
problem Characterizing flat subsystems in flat systems of differential dimension 2.
method Analyzing subsystems of a flat system of differential dimension at most 2.
result Flat subsystems of a flat system of differential dimension at most 2 exist and can have independent time-uniform outputs.
Study flat manifolds and their collapse using Teichmüller theory.
problem Understanding the collapse of flat manifolds and orbifolds.
method Algebraic description of Teichmüller and moduli spaces, study of boundaries.
result Every closed flat orbifold can be obtained by collapsing closed flat manifolds, and collapsed limits of 3-manifolds are classified.
Two conjectures on Ricci-flat metrics verified under specific conditions.
problem Properties of Ricci-flat metrics on complex manifolds.
method Analyzing conjectures and verifying them under specific conditions.
result Conjectures verified for certain types of metrics on complex surfaces.
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Classifies spin structures on 4D almost-flat manifolds.
problem Determining spin structures on almost-flat manifolds.
method Utilised the canonical orthogonal representation of fundamental groups.
result 15 out of 127 orientable families are non-spin.
Flatness of manifolds with open flat subsets proven using bipolar comparisons.
problem Conditions for flatness in Riemannian manifolds.
method Using (3,3)-bipolar comparisons and open flat subsets.
result Flatness of manifolds proven under specific conditions.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
New examples of special geometric solitons found.
problem Finding new types of geometric solitons.
method Constructing specific examples of Bach-flat gradient Ricci solitons.
result Examples of solitons that are neither conformally flat nor Einstein.