arXiv subjects

Hans-Joachim Hein

Publications and source records attributed to Hans-Joachim Hein.

At least 19 recordsLinked to original sources

arXiv2026

Liouville theorems and Evans-Krylov estimates

A classical idea in analysis going back at least to work of Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder type regularity estimates. The goal of this course is to describe some recent developments of this idea concerning the regularity of the complex Monge-Ampère equation with respect to singular reference metrics. We will start with a quick look at the classical $C^2$ and $C^3$ estimates of Calabi-Aubin-Yau and then present a new proof of the Evans-Krylov $C^{2,α}$ estimate on a Euclidean ball. Based on this we will consider the case of singular backgrounds such as cylinders and cones, discussing some recent work by Hein, Tosatti, Lee and Klemmensen. Our discussion is far from complete and knowledge of Kähler geometry including the Aubin-Yau theorems is assumed. The course includes five exercises with solutions and a problem list.

math.DG↗

arXiv2025

Collapsing immortal Kähler-Ricci flows

We consider the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.

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arXiv2025

A continuous cusp closing process for negative Kähler-Einstein metrics

We give an example of a family of smooth complex algebraic surfaces of degree $6$ in $\mathbb{CP}^3$ developing an isolated elliptic singularity. We show via a gluing construction that the unique Kähler-Einstein metrics of Ricci curvature $-1$ on these sextics develop a complex hyperbolic cusp in the limit, and that near the tip of the forming cusp a Tian-Yau gravitational instanton bubbles off.

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arXiv2024

Smooth asymptotics for collapsing Calabi-Yau metrics

We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose Kahler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with k-th order remainders that satisfy uniform C^k-estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for k=0 known from previous work of the second-named author. For k>0 the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

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arXiv2023

Asymptotically Calabi metrics and weak Fano manifolds

We show that any asymptotically Calabi manifold which is Calabi-Yau can be compactified complex analytically to a weak Fano manifold. Furthermore, the Calabi-Yau structure arises from a generalized Tian-Yau construction on the compactification, and we prove a strong uniqueness theorem. We also give an application of this result to the surface case.

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arXiv2022

Classification of asymptotically conical Calabi-Yau manifolds

A Riemannian cone $(C, g_C)$ is by definition a warped product $C = \mathbb{R}^+ \times L$ with metric $g_C = dr^2 \oplus r^2 g_L$, where $(L,g_L)$ is a compact Riemannian manifold without boundary. We say that $C$ is a Calabi-Yau cone if $g_C$ is a Ricci-flat Kähler metric and if $C$ admits a $g_C$-parallel holomorphic volume form; this is equivalent to the cross-section $(L,g_L)$ being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler $4$-manifolds without twistor theory.

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arXiv2021

Asymptotics of Kähler-Einstein metrics on complex hyperbolic cusps

Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat Kähler metric on $D$. Then $h$ is unique up to scaling, and, for some closed tubular neighborhood $V$ of the zero section $D \subset L$, the form $ω_h = -(n+1)i\partial\overline\partial\log(-{\log h})$ defines a complete Kähler-Einstein metric on $V \setminus D$ with ${\rm Ric}(ω_h) = -ω_h$. In fact, $ω_h$ is complex hyperbolic, i.e., the holomorphic sectional curvature of $ω_h$ is constant, and $ω_h$ has the usual doubly-warped cusp structure familiar from complex hyperbolic geometry. In this paper, we prove that if $U$ is another closed tubular neighborhood of the zero section and if $ω$ is a complete Kähler-Einstein metric with ${\rm Ric}(ω) = -ω$ on $U \setminus D$, then there exist a Hermitian metric $h$ as above and a $δ\in \mathbb{R}^+$ such that $ω- ω_{h} = O(e^{-δ\sqrt{-{\log h}}})$ to all orders with respect to $ω_h$ as $h \to 0$. This rate is doubly exponential in the distance from a fixed point, and is sharp.

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arXiv2021

The renormalized volume of a 4-dimensional Ricci-flat ALE space

We introduce a natural definition of the renormalized volume of a 4-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of 4-dimensional Ricci-flat ALE spaces are Kronheimer's gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer's period map.

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arXiv2020

Higher-order estimates for collapsing Calabi-Yau metrics

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform C^infinity estimate. We then apply these local results to the case of collapsing Calabi-Yau metrics on compact Calabi-Yau manifolds. In this global setting, the C^0 estimate required as a hypothesis in our new local C^alpha and C^infinity estimates is known to hold thanks to earlier work of the second-named author.

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arXiv2019

On the classification of ALE Kähler manifolds

The underlying complex structure of an ALE Kähler manifold is exhibited as a resolution of a deformation of an isolated quotient singularity. As a consequence, there exist only finitely many diffeomorphism types of minimal ALE Kähler surfaces with a given group at infinity.

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arXiv2018

Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.

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arXiv2017

A Liouville theorem for the complex Monge-Ampère equation on product manifolds

Let $Y$ be a closed Calabi-Yau manifold. Let $ω$ be the Kähler form of a Ricci-flat Kähler metric on $\mathbb{C}^m \times Y$. We prove that if $ω$ is uniformly bounded above and below by constant multiples of $ω_{\mathbb{C}^m} + ω_Y$, where $ω_{\mathbb{C}^m}$ is the standard flat Kähler form on $\mathbb{C}^m$ and $ω_Y$ is any Kähler form on $Y$, then $ω$ is actually equal to a product Kähler form, up to a certain automorphism of $\mathbb{C}^m \times Y$.

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arXiv2016

Calabi-Yau manifolds with isolated conical singularities

Let $X$ be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let $L$ be an ample line bundle on $X$. Assume that the pair $(X,L)$ is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point $x \in X$ there exist a Kahler-Einstein Fano manifold $Z$ and a positive integer $q$ dividing $K_Z$ such that $-\frac{1}{q}K_Z$ is very ample and such that the germ $(X,x)$ is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of $\frac{1}{q}K_{Z}$. We prove that up to biholomorphism, the unique weak Ricci-flat Kahler metric representing $2πc_1(L)$ on $X$ is asymptotic at a polynomial rate near $x$ to the natural Ricci-flat Kahler cone metric on $\frac{1}{q}K_Z$ constructed using the Calabi ansatz. In particular, our result applies if $(X, \mathcal{O}(1))$ is a nodal quintic threefold in $\mathbb{P}^4$. This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.

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arXiv2016

Mass in Kähler Geometry

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.

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arXiv2015

Remarks on the collapsing of torus fibered Calabi-Yau manifolds

One of the main results of the paper arXiv:1108.0967 by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kahler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and simplify some of the underlying analysis.

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arXiv2014

Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler-Einstein, then, applying results from our previous paper, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-ε}) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce new methods to prove an analogous statement (with rate O(r^{-0.0128})) when X = Bl_{p}P^3 and D = Bl_{p_1,p_2}P^2 is the strict transform of a smooth quadric through p in P^3. Here D is no longer Kahler-Einstein, but the normal S^1-bundle to D in X admits an irregular Sasaki-Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi-Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.

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arXiv2014

Asymptotically cylindrical Calabi-Yau manifolds

Let $M$ be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to $[0,\infty) \times X$ for some compact connected Ricci-flat manifold $X$. We begin by proving general structure theorems for $M$; in particular we show that there is no loss of generality in assuming that $M$ is simply-connected and irreducible with Hol$(M)$ $=$ SU$(n)$, where $n$ is the complex dimension of $M$. If $n > 2$ we then show that there exists a projective orbifold $\bar{M}$ and a divisor $\bar{D}$ in $|{-K_{\bar{M}}}|$ with torsion normal bundle such that $M$ is biholomorphic to $\bar{M}\setminus\bar{D}$, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where $\bar{M}$ is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair $(\bar{M}, \bar{D})$ we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on $\bar{M}\setminus\bar{D}$.

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arXiv2014

Asymptotically conical Calabi-Yau manifolds, III

In a recent preprint, Chi Li proved that aymptotically conical complex manifolds with regular tangent cone at infinity admit holomorphic compactifications (his result easily extends to the quasiregular case). In this short note, we show that if the open manifold is Calabi-Yau, then Chi Li's compactification is projective algebraic. This has two applications. First, every Calabi-Yau manifold of this kind can be constructed using our refined Tian-Yau type theorem from the second article in this series. Secondly, we prove classification theorems for such manifolds via deformation to the normal cone. This includes Kronheimer's classification of ALE spaces and a uniqueness theorem for Stenzel's metric.

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