arXiv research

New examples of Ricci limit spaces with mixed tangent cones.

Unique cylindrical tangent cone for Simons' hypersurface found.

Analytic sets with unique infinite tangent cone are algebraic.

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 …

Uniqueness proven for stable hypersurface tangent cones.

Uniqueness proven for cylindrical tangent cones in high dimensions.

New Calabi-Yau metrics found on complex symmetric spaces.

Characterizes tangent cones for specific connections on reflexive sheaves.

Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

Kähler-Ricci flows' tangent cones are algebraic varieties.

Unique tangent cones found for boundary points of 2D almost-minimizing currents.

Paper proves unique tangent maps for complex maps into algebraic varieties.

Study shows unique tangent cones for area-minimizing currents at boundary points.

Study of tangent cones at infinity for algebraic sets.

Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.

New special Lagrangian submanifolds with cylindrical tangent cones are constructed.

New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.

Generalizes Connes's tangent groupoid for sub-Riemannian geometry.

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

Consider a limit space $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α^n$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of $Y$ at a point $p\in Y$ are known to be metric cones $C(X)$, however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the close…

Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.

We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].

The paper constructs 4-manifolds with nonnegative Ricci curvature and specific asymptotic cones.

Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of…

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform r…

The study examines singularities in flows with curvature bounds and identifies unique tangent flows.

Unique tangent cones found for Kahler-Einstein metrics on singular varieties.

Study on Kähler-Einstein metrics with polynomial convergence rates.

Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.

Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.

We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on $μ$-polystable holomorphic bundles over $\mathbf{P}^{n-1}$.

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this …

New Calabi-Yau metrics with conical singularities are created near complex lines.

Geodesic flow mixing on convex projective manifolds proven.

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…

If M is a smooth compact Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. If S is an embedded submanifold of M, and $μ$ is an absolutely continuous measure on S, then we compute the tangent cone of P(M) at $μ$.

We consider the Calabi-Yau metrics on $\mathbf{C}^n$ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone $\mathbf{C}\times A_1$ at infinity for the $(n-1)$-dimensional Stenzel cone $A_1$. We show that up to scaling and isometry this Calabi-Yau metric on $\mathbf{C}^n$ is unique. We al…

We prove some epsilon regularity results for n-dimensional minimal two-valued Lipschitz graphs. The main theorems imply uniqueness of tangent cones and regularity of the singular set in a neighbourhood of any point at which at least one tangent cone is equal to a pair of transversely intersecting multiplicity one n-dim…

A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic volume ratio of gradient Ricci solitons.

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be prod…

In this paper we study the analytic tangent cones of admissible Hermitian-Yang-Mills connections near a homogeneous singularity of a reflexive sheaf, and relate it to the Harder-Narasimhan-Seshadri filtration. We also give an algebro-geometric characterization of the bubbling set. This strengthens our previous result.

We construct infinitely many complete Calabi-Yau metrics on $\mathbf{C}^n$ for $n \geq 3$, with maximal volume growth, and singular tangent cones at infinity. In addition we construct Calabi-Yau metrics in neighborhoods of certain isolated singularities whose tangent cones have singular cross section, generalizing work…

In this short note, we would like to give a construction of parallel transport for tangent cones lying in the interior of a geodesic in Wasserstein space. We give a complete proof for the linear part of the tangent space, and show that a construction for the full tangent cones follows from some natural lemmas on Wasser…

Study limits of manifolds with Kato bound Ricci curvature, proving volume convergence.