A detailed look at the Calabi-Eguchi-Hanson spaces
Jørgen Olsen Lye
备注：
22 pages. The second version includes the following changes: Added a short result about the holomorphic volume form (Prop. 8), amended an important gap in the proof of Thm. 9, added a description of local coordinates on the canonical line bundle and described the metric in these coordinates (Eq. 22)
This article takes a detailed look at the Ricci-flat metrics introduced by Eguchi-Hanson and Calabi on the canonical line bundle of complex projective space. We give a description of these spaces as resolutions of certain orbifold singularities. We then compute the curvature explicitly and show that all compact, minimal submanifolds are contained in the zero section. This extends a result by Tsai and Wang.
Let $g_{EH}$ be the Eguchi-Hanson metric on the blow-up of $\mathbb{C}^2$ at the origin. In this paper we show that $mg_{EH}$ is not balanced for any positive integer $m$.
We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 /\mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R}…
We construct two types of Eguchi-Hanson metrics with the negative constant scalar curvature. The type I metrics are Kähler. The type II metrics are ALH whose total energy can be negative.
We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled S…
We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $ι$ preserving the $G_2$-structure. Then $M/{\langle ι\rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $λ$ on $L$. Much of…
In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if…
We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G_2-ma…
We find new obstructions to the desingularization of compact Einstein orbifolds by smooth Einstein metrics. These new obstructions, specific to the compact situation, raise the question of whether a compact Einstein $4$-orbifold which is limit of Einstein metrics bubbling out Eguchi-Hanson metrics has to be Kähler. We then test these obstructions to discuss if it is possible to produce a Ricci-flat but not Kähler metric by the most promising desingularization configuration proposed by Page in 1981. We identify $84$ obstructions which, once compared to the $57$ degrees of freedom, indicate that…
We give a description of all $G$-invariant Ricci-flat Kähler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As the simplest application, we obtain the explicit description of the set of all complete $\mathrm{SO}(3)$-invariant Ricci-flat Kähler metrics on $T{\mathbb S}^2$, which includes the well-known Eguchi-Hanson-Stenzel metrics and a new one-parameter family of metrics.
We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.
Consider an Einstein orbifold $(M_0,g_0)$ of real dimension $2n$ having a singularity with orbifold group the cyclic group of order $n$ in ${\rm{SU}}(n)$ which is generated by an $n$th root of unity times the identity. Existence of a Ricci-flat Kähler ALE metric with this group at infinity was shown by Calabi. There is a natural "approximate" Einstein metric on the desingularization of $M_0$ obtained by replacing a small neighborhood of the singular point of the orbifold with a scaled and truncated Calabi metric. In this paper, we identify the first obstruction to perturbing this approximate E…
The Kummer Construction of Calabi-Yau and Hyper-Kähler Metrics on the $K3$ Surface, and Large Families of Volume Non-collapsed Limiting Compact Hyper-Kähler Orbifolds
