Gábor Székelyhidi

Professor
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208
email : gaborsz at northwestern.edu

I am interested in geometric analysis and complex differential geometry. Much of my work is motivated by trying to find canonical metrics, such as extremal or Kähler-Einstein metrics on projective manifolds.

Special Metrics in Complex Geometry, May 16--20, 2022

Notre Dame Geometry & Topology RTG

Northwestern, UIC Complex Geometry Seminar

Center for Mathematics Thematic Program on Kähler geometry, June 12 - 30, 2017

Notre Dame Geometric Analysis Seminar, Past years

MMP and Ricci flow learning seminar

Teaching

Papers, CV

• (with Y. Chen, S.-K. Chiu, M. Hallgren, T. D. Tô, F. Tong) On Kähler-Einstein Currents
  [abstract] [pdf]
  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 Lp for p>(2n−1)/n, then the metric defines an RCD space.

• (with Y. Li) Singularity formations in Lagrangian mean curvature flow
  [abstract] [pdf]
  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.

• Singular Kähler-Einstein metrics and RCD spaces
  [abstract] [pdf]
  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.

• Recent progress on minimal hypersurfaces with cylindrical tangent cones
  [abstract] Aequationes Math. 97 (2023), no. 5-6, 1083-1106.
  We survey some recent results on minimal hypersurfaces in Rⁿ⁺¹ with cylindrical tangent cones. We discuss the question of the uniqueness of tangent cones, the behavior of certain minimal hypersurfaces with cylindrical tangent cones, and a Liouville type theorem for entire minimal hypersurfaces.

• (with N. Edelen) A Liouville-type theorem for cylindrical cones
  [abstract] [pdf] Comm. Pure Appl. Math. 77 (2024), no. 8, 3557–3580.
  Suppose that C₀ ⊂ Rⁿ⁺¹ is a smooth strictly minimizing and strictly stable minimal hypercone, l≥0, and M a complete embedded minimal hypersurface of R^n+1+l lying to one side of C=C₀x R^l. If the density at infinity of M is less than twice the density of C, then we show that M=H(λ) x R^l, where {H(λ)}λ is the Hardt-Simon foliation of C₀. This extends a result of L. Simon, where an additional smallness assumption is required for the normal vector of M.

• (with J. D. Lotay, F. Schulze) Neck pinches along the Lagrangian mean curvature flow of surfaces
  [abstract] [pdf]
  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 L₀ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.

• (with J. D. Lotay, F. Schulze) Ancient solutions and translators of Lagrangian mean curvature flow
  [abstract] [pdf] to appear in Publ. Math. IHES
  Suppose that M is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in Cⁿ. We show that if M has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then M is a translator. In particular in C², all zero-Maslov, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.

• (with S.-K. Chiu) Higher regularity for singular Kähler-Einstein metrics
  [abstract] [pdf] Duke Math. J. 172 (2023), no. 18, 3521–3558.
  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 ∂∂-exact Calabi-Yau metrics with maximal volume growth. This generalizes a result of Conlon-Hein, which applies to the case of asymptotically conical manifolds.

• Minimal hypersurfaces with cylindrical tangent cones
  [abstract] [pdf]
  First we construct minimal hypersurfaces M⊂Rⁿ⁺¹ in a neighborhood of the origin, with an isolated singularity but cylindrical tangent cone C×R, for any strictly minimizing strictly stable cone C in Rⁿ. 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×R at infinite order, then V=C×R in a neighborhood of the origin. Using this we show that for quadratic cones C=C(S^p×S^q), in dimensions n>8, all O(p+1)×O(q+1)-invariant minimal hypersurfaces with tangent cone C×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×R or has an isolated singularity at the origin.

• Uniqueness of certain cylindrical tangent cones
  [abstract] [pdf]
  We show that the cylindrical tangent cone C×R for an area-minimizing hypersurface is unique, where C is the Simons cone C_S=C(S³×S³). 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×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.

• (with B. Weinkove) Weak Harnack inequalities for eigenvalues and constant rank theorems
  [abstract] [pdf] Comm. Partial Differential Equations 46 (2021), no. 8, 1585--1600
  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.

• (with J. Ross) Twisted Kähler-Einstein metrics
  [abstract] [pdf] Pure Appl. Math. Q. 17 (2021), no. 3, 1025–1044.

  We prove an existence result for twisted Kähler-Einstein metrics, assuming an appropriate twisted K-stability condition. An improvement over earlier results is that certain non-negative twisting forms are allowed

• Uniqueness of some Calabi-Yau metrics on Cⁿ
  [abstract] [pdf] Geom. Funct. Anal. 30 (2020), no. 4, 1152--1182.

  We consider the Calabi-Yau metrics on Cⁿ constructed recently by Yang Li, Conlon-Rochon, and the author, that have tangent cone C×A₁ at infinity for the (n−1)-dimensional Stenzel cone A₁. We show that up to scaling and isometry this Calabi-Yau metric on Cⁿ is unique. We also discuss possible generalizations to other manifolds and tangent cones.

• (with G. Liu) Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below, II
  [abstract] [pdf] Comm. Pure Appl. Math. 74 (2021), no. 5, 909–931.

  We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curvature bounds.

• (with G. Liu) Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below
  [abstract] [pdf] Geom. Funct. Anal. 32 (2022), no. 2, 236–279.

  A fundamental result of Donaldson-Sun states that non-collapsed Gromov-Hausdorff limits of polarized Kähler manifolds, with 2-sided Ricci curvature bounds, are normal projective varieties. We extend their approach to the setting where only a lower bound for the Ricci curvature is assumed. More precisely, we show that non-collapsed Gromov-Hausdorff limits of polarized Kähler manifolds, with Ricci curvature bounded below, are normal projective varieties. In addition the metric singularities are precisely given by a countable union of analytic subvarieties.

• (with M. Gursky) A local existence result for Poincaré-Einstein metrics
  [abstract] [pdf] Adv. Math. 361 (2020), 106912, 50 pp.

  Given a closed Riemannian manifold (M,g_M) of dimension n ≥ 3, we prove the existence of a conformally compact Einstein metric g₊ defined on a collar neighborhood M x (0,1] whose conformal infinity is [g_M].

• Kähler-Einstein metrics
  [abstract] [pdf] Proc. Sympos. Pure Math., 99., 331--361

  We survey the theory of Kähler-Einstein metrics, with particular focus on the circle of ideas surrounding the Yau-Tian-Donaldson conjecture for Fano manifolds.

• Degenerations of Cⁿ and Calabi-Yau metrics
  [abstract] [pdf] Duke Math. J. 168 (2019), no. 14, 2651--2700

  We construct infinitely many complete Calabi-Yau metrics on Cⁿ for n≥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 of Hein-Naber.

• (with R. Dervan) The Kähler-Ricci flow and optimal degenerations
  [abstract] [pdf] J. Differential Geom. 116 (2020), no. 1, 187–203.

  We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" special degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He.
  
  We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a Kähler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.

• (with R. Seyyedali) Extremal metrics on blowups along submanifolds
  [abstract] [pdf] J. Differential Geom. 114 (2020), no. 1, 171--192

  We give conditions under which the blowup of an extremal Kähler manifold along a submanifold of codimension greater than two admits an extremal metric. This generalizes work of Arezzo-Pacard-Singer, who considered blowups in points.

• (with T. Collins) Sasaki-Einstein metrics and K-stability
  [abstract] [pdf] Geom. Topol. 23 (2019), no. 3, 1339--1413.

  We show that a polarized affine variety admits a Ricci flat Kähler cone metric, if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.

• (with B. Weinkove) On a constant rank theorem for nonlinear elliptic PDEs
  [abstract] [pdf] Discrete Contin. Dyn. Syst. Ser. A, 36 (2016) no. 11, 6523--6532

  We give a new proof of Bian-Guan's constant rank theorem for nonlinear elliptic equations. Our approach is to use a linear expression of the eigenvalues of the Hessian instead of quotients of elementary symmetric functions.

• (with V. Datar) Kähler-Einstein metrics along the smooth continuity method
  [abstract] [pdf] Geom. Funct. Anal. 26 (2016) no. 4, 975--1010

  We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kähler-Ricci solitons.

• (with V. Tosatti, B. Weinkove) Gauduchon metrics with prescribed volume form
  [abstract] [pdf] Acta Math. 219 (2017), no. 1, 181--211

  We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

• Fully non-linear elliptic equations on compact Hermitian manifolds
  [abstract] [pdf] J. Differential Geom. 109 (2018), no. 2, 337--378

  We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kähler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.

• (with T. Collins) Convergence of the J-flow on toric manifolds
  [abstract] [pdf] J. Differential Geom. 107 (2017) no. 1, 47--81

  We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing results on more general inverse sigma_k equations on Kahler manifolds.

• Extremal Kähler metrics
  [abstract] [pdf] Proceedings of the ICM, 2014

  This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then turn to the question of what one expects when no extremal metric exists.

• The partial C⁰-estimate along the continuity method
  [abstract] [pdf] J. Amer. Math. Soc. 29 (2016), 537--560

  We prove that the partial C⁰-estimate holds for metrics along Aubin's continuity method for finding Kähler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on the analogous problem for conical Kähler-Einstein metrics.

• (with M. Lejmi) The J-flow and stability
  [abstract] [pdf] Adv. Math. 274 (2015), 404--431

  We study the J-flow from the point of view of an algebro-geometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang-Lai is reflected by the optimal destabilizer. Finally we prove a general existence result on complex tori.

• Blowing up extremal Kähler manifolds II
  [abstract] [pdf] Invent. Math. 200 (2015), no. 3, 925--977

  This is a continuation of the work of Arezzo-Pacard-Singer and the author on blowups of extremal Kähler manifolds. We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. As an application we prove that if a Kähler manifold M of dimension greater than 2 admits a cscK metric, then the blowup of M at a point admits a cscK metric if and only if it is K-stable, as long as the exceptional divisor is sufficiently small.

• A remark on conical Kähler-Einstein metrics
  [abstract] [pdf] Math. Res. Lett. 20 (2013) n. 3., 581--590

  We give some non-existence results for Kähler-Einstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy from the point of view of log K-stability.

• Remark on the Calabi flow with bounded curvature
  [abstract] [pdf] Univ. Iagel. Acta Math. 50 (2013), 107--115

  In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.

• (with T. Collins) The twisted Kähler-Ricci flow
  [abstract] [pdf] J. Reine Angew. Math. 716 (2016), 179--205

  In this paper we study a generalization of the Kähler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kähler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kähler-Ricci flow, and it builds on work of Tian-Zhu.

• (with T. Collins) K-Semistability for irregular Sasakian manifolds
  [abstract] [pdf] J. Differential Geom. 109 (2018), no. 1, 81--109

  We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.

• Filtrations and test-configurations
  [abstract] [pdf] Math. Ann. 362 (2015), 451--484

  We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We make some progress towards proving that if a manifold with no automorphisms admits a cscK metric, then it satisfies this stronger stability notion. Finally we discuss the relation with the birational transformations in the definition of b-stability.

• (with J. Song and B. Weinkove) The Kähler-Ricci flow on projective bundles
  [abstract] [pdf] Int. Math. Res. Not. 2013, 243--257

  We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.

• (with D. McFeron) On the positive mass theorem for manifolds with corners
  [abstract] [pdf] Comm. Math. Phys. 313 (2012), 425--443

  We study the positive mass theorem for certain non-smooth metrics following P. Miao's work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

• (with Renjie Feng) Periodic solutions of Abreu's equation
  [abstract] [pdf] Math. Res. Lett. 18 (2011) n. 6., 1271--1279

  We solve Abreu's equation with periodic right hand side, in any dimension. This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety.

• On blowing up extremal Kähler manifolds
  [abstract] [pdf] Duke Math. J., 161 (2012) n. 8, 1411--1453

  We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-Singer. We also study the K-polystability of these blowups, sharpening a result of Stoppa in this case. As an application we show that the blowup of a Kahler-Einstein manifold at a point admits a constant scalar curvature Kahler metric in classes that make the exceptional divisor small, if it is K-polystable with respect to these classes.

• (with J. Stoppa) Relative K-stability of extremal metrics
  [abstract] [pdf] J. Eur. Math. Soc. 13 (2011) n. 4, 899--909

  We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.

• (with V. Tosatti) Regularity of weak solutions of a complex Monge-Ampère equation
  [abstract] [pdf] Analysis & PDE 4 (2011), n. 3, 369--378

  We prove the smoothness of weak solutions to an elliptic complex Monge-Ampère equation, using the smoothing property of the corresponding parabolic flow.

• (with O. Munteanu) On convergence of the Kähler-Ricci flow
  [abstract] [pdf] Comm. Anal. Geom. 19 (2011), n. 5, 887--904

  We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then give different situations in which the condition on the Mabuchi energy holds.

• Greatest lower bounds on the Ricci curvature of Fano manifolds
  [abstract] [pdf] Compositio Math. 147 (2011), 319--331

  On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding Kähler-Einstein metrics. We show that on P^2 blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

• The Kähler-Ricci flow and K-polystability
  [abstract] [pdf] Amer. J. Math. 132 (2010), 1077--1090

  We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small perturbation of a cscK manifold admits a cscK metric if it is K-polystable.

• The Calabi functional on a ruled surface
  [abstract] [pdf] Ann. Sci. Éc. Norm. Supér. 42 (2009), 837--856

  We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.

• Optimal test-configurations for toric varieties
  [abstract] [pdf] J. Differential Geom. 80 (2008), 501--523

  On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists for all time on a toric variety then it minimises the Calabi functional. In this case the infimum of the Calabi functional is given by the supremum of the normalised Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.

• Extremal metrics and K-stability
  [abstract] [pdf] Bull. London Math. Soc. 39 (2007), 76--84

  We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kähler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

• (with M. Laczkovich) Harmonic analysis on discrete Abelian groups
  [abstract] [link] Proc. Amer. Math. Soc. 133 (2005), 1581--1586
  Let G be an Abelian group and let C^G denote the linear space of all complex-valued functions defined on G equipped with the product topology. We prove that the following are equivalent. (i) Every nonzero translation invariant closed subspace of C^G contains an exponential; that is, a nonzero multiplicative function. (ii) The torsion free rank of G is less than the continuum.

Book

Thesis

Links

• Great Lakes Geometry Conference 2014 - website
• Northwestern Math Department
• Notre Dame Math Department
• Geometry at Imperial
• Columbia Math Department
• László Székelyhidi Jr.
• Sonja Mapes Székelyhidi