Classifies tangent cones of Kähler metrics with cone singularities.
problem Understanding the behavior of Kähler metrics near singular points.
method Constructs and classifies tangent cones for non-collapsed sequences of Kähler-Einstein metrics.
result Classifies possible tangent cones in two complex dimensions.
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
Unique Calabi-Yau metrics found on C^n.
problem Finding unique Calabi-Yau metrics on C^n.
method Analyzing metrics with specific cone structures.
result Calabi-Yau metrics on C^n are unique up to scaling and isometry.
New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.
problem Uncertainty in homeomorphism types of tangent cones of non-collapsed Ricci limit spaces.
method Construction of limit spaces in all dimensions at least 5.
result Any finite collection of manifolds can appear as cross sections of tangent cones of the same point.
New Calabi-Yau metrics found on C^n for n>=3.
problem Finding Calabi-Yau metrics on complex manifolds.
method Constructing metrics with maximal volume growth and singular tangent cones.
result Infinitely many complete Calabi-Yau metrics on C^n.
Study on Kähler-Einstein metrics with polynomial convergence rates.
problem Understanding convergence rates of singular Kähler-Einstein metrics.
method Analyzing non-collapsed limits and tangent cones of polarized Kähler-Einstein manifolds.
result Polynomial convergence of Kähler potentials on tangent cones.
The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.
problem Preventing stable tangent cones for area-minimizing boundaries under specific metrics.
method Developed a perturbation theorem and used spectral theory and compactness arguments.
result A residual set of metrics on Sn+1 precludes linearly stable tangent cones for area-minimizing boundaries. Generalizes Connes's tangent groupoid for sub-Riemannian geometry.
problem Calculating tangent cones in sub-Riemannian geometry.
method Constructs a completion of MimesMimesR+imes using sub-Riemannian metric. result Calculates all tangent cones in Gromov-Hausdorff distance.
Unique tangent cones found for Kahler-Einstein metrics on singular varieties.
problem Finding unique tangent cones for Kahler-Einstein metrics on singular varieties.
method Analyzing unique Ricci flat currents with local bounded potential.
result Local tangent cones of the unique Kahler-Einstein metric are unique.
New Calabi-Yau metrics with conical singularities are created near complex lines.
problem Creating Calabi-Yau metrics with conical singularities near complex lines.
method Using branched covering arguments to construct metrics with conical singularities.
result Calabi-Yau metrics with unstable conical singularities are successfully created.
Kähler-Ricci flows' tangent cones are algebraic varieties.
problem Understanding the structure of Kähler-Ricci flows' tangent cones.
method Analyzing tangent cones as normal affine algebraic varieties and using Hörmander's L2 estimate. result The regular set of tangent cones coincides with the algebraic regular set.
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
Complete Calabi-Yau metrics found on higher-dimensional complex space.
problem Finding complete Calabi-Yau metrics in higher dimensions.
method Constructing metrics with specific properties on Cn. result Examples of complete Calabi-Yau metrics with Euclidean volume growth.
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Consider a limit space (Mα,gα,pα)→GH(Y,dY,p), where the Mαn have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of Y at a point p∈Y are known to be metric cones C(X), however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the close…
New complete Calabi-Yau metric on C^3 with maximal volume growth.
problem Modeling collapsing Calabi-Yau threefolds near nodal points.
method Existence result perturbed into an actual solution with correction of slowly decaying error terms.
result Complete Calabi-Yau metric on C^3 with maximal volume growth and non-standard geometry near singularity.
Study on higher-dimensional quasigeodesics in metric spaces.
problem Understanding asymptotic structure of Morse quasiflats.
method Proving asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity.
result Morse quasiflats exhibit Euclidean volume growth rigidity.
Study shows finitely generated fundamental groups for certain nonnegative Ricci curvature manifolds.
problem Understanding fundamental groups of manifolds with nonnegative Ricci curvature.
method Analyzing tangent cones at infinity and their Euclidean dimensions.
result Fundamental groups are finitely generated under specific curvature and tangent cone conditions.
An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating…
Characterizes minimizing curves in Riemannian manifolds.
problem Finding optimal paths in curved spaces.
method Characterization of prox-regular sets and tangent cones.
result Necessary condition for minimizing curves in prox-regular sets.
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent …
We show that the tangent cone at the identity is not a complete quasiconformal invariant for sub-Riemannian nilpotent groups. Namely, we show that there exists a nilpotent Lie group equipped with left invariant sub-Riemannian metric that is not locally quasiconformally equivalent to its tangent cone at the identity. In…
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
We show that if X is a limit of n-dimensional Riemannian manifolds with Ricci curvature bounded below and γ is a limit geodesic in X then along the interior of γ same scale measure metric tangent cones Tγ(t)X are Hölder continuous with respect to measured Gromov-Hausdorff topology and have the same dimen…
Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.
problem Understanding the asymptotic geometry of ancient Ricci flows with nonnegative Ricci curvature.
method Analyze tangent flows at infinity and use estimates for noncollapsed F-limit metric solitons.
result Two dichotomy theorems for ancient Ricci flows: either the asymptotic volume ratio is zero or every tangent flow is a Ricci flat cone.
Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.
problem Understanding structure of limits of manifolds with Ricci curvature bounds.
method Mosco convergence of Dirichlet energies to Cheeger energy, introduction of monotone quantities, volume convergence.
result Volume convergence to Hausdorff n-measure in limits of manifolds.
Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones Cp,qimesR in R9. Characterizes tangent cones for specific connections on reflexive sheaves.
problem Analyzing tangent cones of admissible Hermitian-Yang-Mills connections over reflexive sheaves.
method Algebro-geometric characterization of analytic tangent cones.
result Complete characterization of tangent cones for admissible Hermitian-Yang-Mills connections over reflexive sheaves.
Let X be the Gromov-Hausdorff limit of a sequence of pointed complete Kähler manifolds (Min,pi) satisfying Ric(Mi)≥−(n−1) and the volume is noncollapsed. We prove that, there exists a Lie group isomorphic to R, acting isometrically, on the tangent cone at each point of X. Moreover, the actio…
The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.
problem Understanding the structure and regularity of Einstein 5-manifolds.
method Analysis of tangent cones, gap theorems for 4-dimensional orbifolds, and careful metric analysis.
result Noncollapsed limits of Einstein 5-manifolds have unique and isolated tangent cones.
Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.
problem Proving uniqueness of cylindrical tangent cones for special Lagrangians.
method Analyzing exact special Lagrangian submanifolds with multiplicity one and cylindrical tangent cones.
result The cylindrical tangent cones are unique under specific conditions.
New methods compute geometry of hyperKähler metrics at infinity.
problem Understanding the geometry of hyperKähler metrics at infinity.
method Quasi-asymptotically conical metrics, Taub-NUT deformations, compactification by manifolds with corners.
result Identifies unique tangent cones and cohomology groups.
The paper proves conditions for C1 regularity of definable sets using tangent cones and paratangent cones.
problem Conditions for C1 regularity of definable sets in o-minimal structures. method Analysis of tangent and paratangent cones to establish C1 regularity. result Equivalence of three conditions for C1 regularity of definable sets. The negative case of the Singular Yamabe Problem concerns the existence and behavior of complete metrics with constant negative scalar curvature on the complement of a closed set in a compact Riemannian manifold which are conformally equivalent to a smooth metric on this compact manifold. When the closed set is a smoot…
Study examines causal properties of Finsler spacetimes with cone Killing vectors.
problem Characterize causality in Finsler spacetimes with specific Killing vectors.
method Explores the relationship between wind Riemannian structures and spacetimes with cone Killing vectors, focusing on Finsler-Kropina metrics.
result Characterizes causality properties using metric-type properties of Finslerian structures.
Developing a non-symmetric strainer theory for spaces with non-negative curvature beyond Alexandrov geometry.
problem Studying spaces with non-Riemannian curvature beyond Alexandrov geometry.
method Introducing a weak quadruple comparison principle and developing a strainer theory.
result Spaces have constant integer dimension, measure contraction property, and unique Banach tangent cones.
Proves uniqueness of tangent cones for specific types of connections.
problem Uniqueness of tangent cones for Hermitian Yang-Mills connections with isolated singularities.
method Simple direct proof using μ-polystable holomorphic bundles over P^n-1.
result Uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections.
New Einstein RCD spaces found with cone singularities.
problem Existence of Einstein RCD spaces with cone singularities.
method Characterization of RCD spaces and cone singularity analysis.
result Existence of smooth non-compact 4-manifolds with ALE Ricci-flat RCD(0,4) metrics.
Proves new results on K-polystability of Q-Fano varieties.
problem K-polystability of Q-Fano varieties.
method Algebraic and differential-geometric arguments.
result Metric tangent cones of K-semistable log Fano cones are K-polystable.
Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.
problem Boundary behavior of the singular Yamabe problem near singular boundaries.
method Analysis of asymptotic behaviors and derivation of optimal estimates for background metrics.
result Solutions are well approximated by solutions in tangent cones at singular points.
Study of compactifications for Kähler-Einstein Fano manifolds.
problem Compactification of moduli spaces of Kähler-Einstein Fano manifolds.
method Geometry of metric tangent cones and algebro-geometric study of singularities.
result First concrete examples of Gromov-Hausdorff compactifications in complex dimensions >2.
Paper proves Hölder continuity of tangent cones in RCD(K,N) spaces.
problem Understanding the geometry of metric measure spaces with curvature-dimension condition.
method Developed a second order interpolation formula for distance function.
result Tangent cones from rescalings are Hölder continuous along geodesics.
New examples of Ricci limit spaces with mixed tangent cones.
problem Constructing Ricci limit spaces with mixed tangent cones.
method For any integers m≥n≥3, construct a Ricci limit space X_{m,n} with specific tangent cones.
result Found a new example of Ricci limit space with mixed tangent cones.
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
Study tangent cones of Hermitian-Yang-Mills connections near singularities.
problem Characterize analytic tangent cones of admissible Hermitian-Yang-Mills connections.
method Analyze the Harder-Narasimhan-Seshadri filtration and relate to bubbling set.
result Strengthened characterization of tangent cones near homogeneous singularities.
Unique tangent cones found for boundary points of 2D almost-minimizing currents.
problem Characterizing boundary points of two-dimensional almost-minimizing currents.
method Combining epiperimetric inequality and almost-monotonicity formula.
result Tangent cones at singular boundary points are unique.