arXiv research

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…

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 …

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

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…

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

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

Study on flat singular points of area-minimizing currents, defining a singularity degree.

Rectifies flat singular points for area-minimizing currents.

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

The study examines complete Kähler manifolds with nonnegative Ricci curvature and discovers rigidity properties.

We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature $κ\in \{-1,0,1\}$ and cone-angles $\leq π$. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem fo…

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

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

The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.

New Einstein RCD spaces found with cone singularities.

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

We proved a uniqueness theorem of tangent connections for a Yang-Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained controls over the rate of the asymptotic convergence of the connection to the tangent connection under assumptions that the connect…

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones a…

Unique cylindrical tangent cone for Simons' hypersurface found.

Analytic sets with unique infinite tangent cone are algebraic.

Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.

Uniqueness proven for stable hypersurface tangent cones.

The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.

We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…

Uniqueness proven for cylindrical tangent cones in high dimensions.

New Calabi-Yau metrics found on complex symmetric spaces.

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and e…

In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds with curvature decay and controlled holonomy. As a…

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

The paper studies Ricci flow with specific curvature and volume constraints, proving convergence and curvature bounds.

This is the first part in a two-part series on complete Calabi-Yau manifolds asymptotic to Riemannian cones at infinity. We begin by proving general existence and uniqueness results. The uniqueness part relaxes the decay condition $O(r^{-n-ε})$ needed in earlier work to $O(r^{-ε})$, relying on some new ideas about harm…

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 prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…

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.

Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on $S\times\mathbb{R}$ is the tangent bundle of the Teichmüller space of $S$, if $S$ is a closed surface. One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relat…

The paper shows that certain manifolds with nonnegative Ricci curvature have finitely generated fundamental groups.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

The study proves a Liouville theorem for certain asymptotically conical Calabi-Yau manifolds.

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