arXiv research

Unique cylindrical tangent cone for Simons' hypersurface found.

Uniqueness proven for cylindrical tangent cones in high dimensions.

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

New special Lagrangian submanifolds with cylindrical tangent cones are constructed.

Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.

Study harmonic functions on submanifolds and their cones.

Proves uniqueness of blowups for forced mean curvature flow.

Study area minimizing currents in Riemannian manifolds, proving unique structure and decay.

The paper studies minimal graphs with bounded 2-dilation in Euclidean space.

Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.

Proves uniqueness of cylindrical tangent flows in mean curvature flow.

Study steady gradient Ricci solitons with cylindrical tangent flows at infinity.

Study shows generic surfaces avoid complex flow patterns.

The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.

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.

New Calabi-Yau metrics found on complex symmetric spaces.

The study confirms a conjecture about critical points of smooth functions.

Resolves conjecture on cylindrical mean curvature flows in all dimensions.

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.

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.

We consider a unit speed curve $α$ in Euclidean $n$-dimensional space $E^n$ and denote the Frenet frame by $\{v_1,...,v_n\}$. We say that $α$ is a cylindrical helix if its tangent vector $v_1$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms of …

Extends a Liouville theorem for stable minimal hypersurfaces.

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…

In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.

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

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In …

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.

The study quantizes ancient flows in cylinders, revealing their asymptotic behavior.

Study on Kähler-Einstein metrics with polynomial convergence rates.

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

Construct locally minimizing $(1,2)$-clusters with prescribed asymptotic geometry.

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