arXiv research

New Calabi-Yau metrics found on complex symmetric spaces.

Study finds volume minimization principle for conical Calabi-Yau structures on horospherical cones.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

Researchers characterize a specific type of projective variety based on its tangents.

Anosov groups in rank ≤3 have unique ergodic horospherical actions.

The paper extends Descartes' circle theorem to n-flower configurations using hyperbolic geometry.

Unique cylindrical tangent cone for Simons' hypersurface found.

New non-existence results for harmonic maps into perturbed cones.

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 …

Geometric correspondence between spinors and horospheres in hyperbolic space.

Uniqueness proven for stable hypersurface tangent cones.

Uniqueness proven for cylindrical tangent cones in high dimensions.

Characterizes tangent cones for specific connections on reflexive sheaves.

The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fun…

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

New examples of Ricci limit spaces with mixed tangent cones.

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

Kähler-Ricci flows' tangent cones are algebraic varieties.

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.

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.

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

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

Study of line bundles on spherical varieties leads to Calabi-Yau metrics.

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].

We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…

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.

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…

Researchers create metrics on hyperbolic space's tangent bundle.