arXiv research

The article introduces a new convergence concept for Lorentzian spaces and applies it to generalized cones.

Paper constructs flows converging to cones and foliations.

Swallowtails form on maxfaces converging to cone-like singularities.

Gradient shrinking solitons from Ricci flows terminating in cones.

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

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

No semistability found for Calabi-Yau metrics near cones.

In this paper, a metric with G$_2$ holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.

Study on Kähler-Einstein metrics with polynomial convergence rates.

The paper studies how surfaces evolve in a cone under a specific flow.

CoNES optimizes blackbox functions using convex optimization and information geometry.

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2π$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angl…

Unique soliton found on resolved cones.

Given a convex cone in the \emph{prescribed} warped product, we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve along the inverse mean curvature flow, then, by using the convexity of …

We use the momentum construction of Calabi to study the conical Kähler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a single point in finite time, or the flow contracts the cone divisor to a single point …

Study shows flow convergence to smooth K-Ricci outside a divisor with cusp singularity.

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece…

Study circular foliations and shear-radius coordinates on hyperbolic cone surfaces.

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

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…

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…

Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-ma…

The paper studies Kähler-Einstein metrics with singularities and their limits.

Addressing a question of Gromov, we give a rate in Pansu's theorem about the convergence in Gromov-Hausdorff metric of a finitely generated nilpotent group equipped with a left-invariant word metric scaled by a factor 1/n towards its asymptotic cone. We show that due to the possible presence of abnormal geodesics in th…

Study limits of Kähler-Einstein metrics with cone singularities on complex projective manifolds.

We study a cutting-plane method for semidefinite optimization problems (SDOs), and supply a proof of the method's convergence, under a boundedness assumption. By relating the method's rate of convergence to an initial outer approximation's diameter, we argue that the method performs well when initialized with a second-…

We consider the deformation theory of asymptotically conical (AC) and of conically singular (CS) $G_2$-manifolds. In the AC case, we show that if the rate of convergence $ν$ to the cone at infinity is generic in a precise sense and lies in the interval $(-4, 0)$, then the moduli space is smooth and we compute its dimen…

This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges toward…

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

It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…

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

The paper proves uniqueness and convergence of asymptotically conical self-shrinkers and self-expanders.

The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.

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…

Minimal surfaces in spheres constructed from symmetry reductions of ODEs.

Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…

In this note, we prove that on a compact Kähler manifold $X$ carrying a smooth divisor $D$ such that $K_X+D$ is ample, the Kähler-Einstein cusp metric is the limit (in a strong sense) of the Kähler-Einstein conic metrics when the cone angle goes to $0$. We further investigate the boundary behavior of those and prove th…

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence $(M_{i}$ of pointed hyperbolic cone-manifolds with topological type $(M,Σ)$, where $M$ is a closed, orientab…

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle $2πβ$ along the divisor, then for any $β'$ sufficiently close to $β$, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein me…

We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and…

Study group actions in metric spaces, proving convergence of lens spaces.

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we cons…

Study convergence of Yamabe flow on singular spaces with positive constant.

This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Krö16]. Up to a small gap in the case of sin-c…

In this paper, we generalize Chow-Luo's combinatorial Ricci flow to inversive distance circle packing setting. Although the solution to the generalized flow may develop singularities in finite time, we can always extend the solution so as it exists for all time and converges exponentially fast. Thus the generalized flo…

In this paper, we study the (normalized) Ricci flow on surfaces with conical singularities. Long time existence is proved for cone angle smaller than $2π$. In this case, convergence results are obtained if the Euler number is nonpositive.

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…