arXiv research

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

Swallowtails form on maxfaces converging to cone-like singularities.

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.

Proves boundedness of log Fano cone singularities with bounded local volumes.

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.

Uniqueness proven for stable hypersurface tangent cones.

Study shortest geodesics on flat cone spheres with conical singularities.

Study strict stability of cones with isolated singularities.

New Calabi-Yau metrics found on complex symmetric spaces.

Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.

Characterizes K-semistability for log Fano cone singularities.

Promotes Poisson deformations to hyperkähler structures.

We develop some foundations for the study of Kahler-Einstein metrics with cone singularities transverse to a divisor. The main goal is a treatment of the deformation of the cone angle.

Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.

We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this…

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As…

Extends Smale's principle to produce minimal graphs with singularities.

Study inequalities for singular values of rectangular matrices.

Characterizes Q-Gorenstein singularities via K-stability.

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

New method fractures hyperbolic manifolds using cone singularities.

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

In two former papers, the authors independently proved that the space of hyperbolic cone-3-manifolds with cone angles less than 2π and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, i.e.…

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

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 $\mathbf{P}^{n-1}$.

Study on minimizing singular capillary cones with stability and instability results.

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as converge…

Study spherical metrics on flat torus with cone singularities.

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].

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

In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…

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

We study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss-Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these y…

In this paper we investigate the singularities of Lagrangian mean curvature flows in $\mathbf{C}^m$ by means of smooth singularity models. Type I singularities can only occur at certain times determined by invariants in the cohomology of the initial data. In the type II case, these smooth singularity models are asympto…

We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $π$: any two such metrics in are connected by a unique left earthquake. The second result is that the space of ``globally hyperbolic'' AdS manifolds with ``parti…

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

Study on sine-cones' spectra and stability under Ricci-de Turck flow.

Recently, the explicit volume formulae for hyperbolic cone-manifolds, whose underlying space is the 3-sphere and the singular set is the knot $4_1$ and the links $5^2_1$ and $6^2_2$, have been obtained by the second named author and his collaborators. In this paper we explicitly find the hyperbolic volume for cone-mani…

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…

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph $Γ$. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than $2π$ on time-like singular segments). We construct examples of such manifolds, d…

Study properties of solutions with singularities in the negative cone.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

Abstract cone operators prove scalar curvature comparisons on singular manifolds.

Study on Kähler-Einstein metrics with polynomial convergence rates.

We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the view point of geometric control theory. In fact we consider the space of singular (or abnormal) paths on a given five dimensional space endowed with a Cartan distribution, which form another five dimensional space with a…

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

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…