arXiv subjects

Gábor Székelyhidi

Publications and source records attributed to Gábor Székelyhidi.

At least 19 recordsLinked to original sources

arXiv2026

Generic mean curvature flow has finitely many singularities

We show that for a generic compact initial surface in $\mathbb{R}^3$, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.

math.DG↗

arXiv2026

Removability of non-isolated singularities for Einstein metrics and RCD spaces

In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $> 3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,α}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.

math.DG↗

arXiv2026

Nondegenerate neck pinches along the mean curvature flow

We show that for generic smooth compact initial surfaces the mean curvature flow in $\mathbb{R}^3$ has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular time are isolated in spacetime. As an application we give a new approach to constructing a mean curvature flow with surgery for smooth compact initial surfaces in $\mathbb{R}^3$.

math.DG↗

arXiv2026

Mean convex flows with surgery

We construct a mean curvature flow with surgery starting with any compact mean convex hypersurface in $\mathbb{R}^{n+1}$, extending previous results of Huisken-Sinestrari, Brendle-Huisken, and Haslhofer-Kleiner for 2-convex flows. In contrast with previous constructions of mean curvature flow with surgery, the topological surgeries are performed by the flow itself through nondegenerate cylindrical singularities. At the same time the flow needs to be slightly adjusted at finitely many smooth times.

math.DG↗

arXiv2026

Generic neck pinch singularities along 2D Lagrangian mean curvature flow

We introduce a notion of nondegenerate neck pinch singularity along the Lagrangian mean curvature flow of surfaces in a Calabi-Yau surface. We show that such singularities can occur, are stable under small perturbations, and any neck pinch singularity can be perturbed to such a nondegenerate singularity near the singular time. Using this we answer some questions raised by Neves and Joyce. We also introduce nondegenerate teardrop singularities and show that these cannot occur for embedded flows.

math.DG↗

arXiv2025

Uniqueness in the Plateau problem near Quadratic cones

We consider minimal hypersurfaces inside the unit ball whose boundary on the sphere is a small perturbation of the link of a minimizing quadratic cone. We show that such minimal surfaces are uniquely determined by their boundary condition. In particular the solutions of the Plateau problem are unique for boundary conditions given by small perturbations of the link of a quadratic cone.

math.DG↗

arXiv2025

Gromov-Hausdorff limits of collapsing Calabi-Yau fibrations

We study Calabi-Yau metrics on a projective manifold in Kähler classes converging to a semiample class given by a fibration. We show that the Gromov-Hausdorff limit of the metrics is homeomorphic to the base of the fibration and in addition the discriminant locus has Hausdorff codimension at least 2. This resolves conjectures of Tosatti.

math.DG↗

arXiv2025

Remarks on Singular Kähler-Einstein Metrics

We study two different natural notions of singular Kähler-Einstein metrics on normal complex varieties. In the setting of singular Ricci flat Kähler cone metrics that arise as non-collapsed limits of sequences of Kähler-Einstein metrics or Kähler-Ricci flows, we show that an a priori weaker notion is equivalent to the stronger one introduced by Eyssidieux-Guedj-Zeriahi, and in particular the underlying variety has log terminal singularities in this case. Our method applies to more general singular Kähler-Einstein spaces as well, assuming that they define RCD spaces.

math.DG↗

arXiv2025

On Kähler-Einstein Currents

We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs, and an analogous result holds for Kähler-Ricci solitons. In addition we show that if a singular Kähler-Einstein metric can be approximated by smooth metrics on a resolution whose Ricci curvature has negative part that is bounded uniformly in $L^p$ for $p > \frac{2n-1}{n}$, then the metric defines an RCD space.

math.DG↗

arXiv2024

Singular Kähler-Einstein metrics and RCD spaces

We study Kähler-Einstein metrics on singular projective varieties. We show that under an approximation property with constant scalar curvature metrics, the metric completion of the smooth part is a non-collapsed RCD space, and is homeomorphic to the original variety.

math.DG↗

arXiv2024

Singularity formations in Lagrangian mean curvature flow

We study singularities along the Lagrangian mean curvature flow with tangent flows given by multiplicity one special Lagrangian cones that are smooth away from the origin. Some results are: uniqueness of all such tangent flows in dimension two; uniqueness in any dimension when the link of the cone is connected; the existence of nontrivial special Lagrangian blowup limits. We also prove a singular version of Imagi-Joyce-dos Santos's uniqueness result of the Lawlor neck. As an application we prove that in any dimension, singularities that admit a tangent flow given by the union of two transverse planes is modeled on shrinking Lawlor necks at suitable scales.

math.DG↗

arXiv2023

Ancient solutions and translators of Lagrangian mean curvature flow

Suppose that $\mathcal{M}$ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in $\mathbb{C}^n$. We show that if $\mathcal{M}$ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then $\mathcal{M}$ is a translator. In particular in $\mathbb{C}^2$, all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.

math.DG↗

arXiv2023

A Liouville-type theorem for cylindrical cones

Suppose that $\mathbf{C}_0^n \subset \mathbb{R}^{n+1}$ is a smooth strictly minimizing and strictly stable minimal hypercone, $l \geq 0$, and $M$ a complete embedded minimal hypersurface of $\mathbb{R}^{n+1+l}$ lying to one side of $\mathbf{C} = \mathbf{C}_0 \times \mathbb{R}^l$. If the density at infinity of $M$ is less than twice the density of $\mathbf{C}$, then we show that $M = H(λ) \times \mathbb{R}^l$, where $\{H(λ)\}_λ$ is the Hardt-Simon foliation of $\mathbf{C}_0$. This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of $M$.

math.DG↗

arXiv2022

Neck pinches along the Lagrangian mean curvature flow of surfaces

Let $L_t$ be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi-Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. Furthermore, if $L_0$ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.

math.DG↗

arXiv2022

Higher regularity for singular Kähler-Einstein metrics

We study singular Kähler-Einstein metrics that are obtained as non-collapsed limits of polarized Kähler-Einstein manifolds. Our main result is that if the metric tangent cone at a point is locally isomorphic to the germ of the singularity, then the metric converges to the metric on its tangent cone at a polynomial rate on the level of Kähler potentials. When the tangent cone at the point has a smooth cross section, then the result implies polynomial convergence of the metric in the usual sense, generalizing a result due to Hein-Sun. We show that a similar result holds even in certain cases where the tangent cone is not locally isomorphic to the germ of the singularity. Finally we prove a rigidity result for complete $\partial\bar\partial$-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds.

math.DG↗

arXiv2021

Minimal hypersurfaces with cylindrical tangent cones

First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable cone $C$ in $\mathbf{R}^n$. We show that many of these hypersurfaces are area minimizing. Next, we prove a strong unique continuation result for minimal hypersurfaces $V$ with such a cylindrical tangent cone, stating that if the blowups of $V$ centered at the origin approach $C\times \mathbf{R}$ at infinite order, then $V = C\times\mathbf{R}$ in a neighborhood of the origin. Using this we show that for quadratic cones $C = C(S^p \times S^q)$, in dimensions $n > 8$, all $O(p+1) \times O(q+1)$-invariant minimal hypersurfaces with tangent cone $C\times \mathbf{R}$ at the origin are graphs over one of the surfaces that we constructed. In particular such an invariant minimal hypersurface is either equal to $C\times \mathbf{R}$ or has an isolated singularity at the origin.

math.DG↗

arXiv2020

Uniqueness of certain cylindrical tangent cones

We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_S= C(S^3\times S^3)$. Previously Simon proved a uniqueness result for cylindrical tangent cones that applies to a large class of cones $C$, however not to the Simons cone. The main new difficulty is that the cylindrical cone $C_S\times \mathbf{R}$ is not integrable, and we need to develop a suitable replacement for Simon's infinite dimensional Lojasiewicz inequality in the setting of tangent cones with non-isolated singularities.

math.DG↗

arXiv2020

Weak Harnack inequalities for eigenvalues and constant rank theorems

We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian-Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.

math.AP↗