RESEARCH IN PAIRS
Complete Kahler-Einstein Metrics and Gravitational Instantons
June 1 – 19, 2015
June 1 – 19, 2015
The proposed research for the three-week ”Research in Pairs” with Prof. Ioana Suvaina (Vanderbilt University), is part of a longer project aimed at classifying all Ricci-flat Kähler metrics on complex surfaces.
In the Trieste Summer School of 2008, Tian made a conjecture that it is possible to completely classify all complete Ricci-flat Kähler metrics on surfaces in terms of the volume growth of the metric: ALE (Euclidean), ALF (cubic), ALG, or ALH.
Suvaina, in [Su], provides a complete and explicit classification of the ALE Ricci-flat Kähler surfaces. This is a generalization of Kronheimer’s results on hyper Kähler manifolds [K], and establishes the correspondence between ALE Ricci-flat Kähler manifolds and a special class of deformations, Q-Gorenstein, of isolated quotient singularities.
Rasdeaconu-Suvaina, in [RS], analyzes the relation between the construc- tion in [Su] and the metrics constructed by Tian and Yau [TY1] [TY2] on the complement of a divisor in a compact surface.
These two results complete the proof of Tian’s conjecture for the ALE case, and provide the relations between all know examples.
During our three-week stay at CIRM, we now propose to attack the case of ALF metrics: relating the Ricci-flat Kähler metrics constructed by Tian and Yau on the complement of a divisor on a compact surface, to the explicit examples of Asymptotically Locally Flat (ALF) metrics: the multi-Taub- NUT metrics of Hawking [H], LeBrun [L], Cherkis-Hitchin [CH].
In the case of ALF hyper Kähler manifolds of cyclic type, Minerbe [M] completed the classification showing that the only possible examples are the trivial R3 × S1, or the multi-Taub-NUT metrics. Such a classification is unknown if one considers the more general case of ALF Rici-flat Kähler surfaces (not necessary hyper kähler), or if the topological structure at infinity is not specified. For example, in [Su] examples of ALF Ricci-flat Kähler surfaces with cyclic fundamental group at infinity are presented. They are not of cyclic type, and are obtained as quotients of the hyper kähler cases.
In Biquard-Minerbe [BM], a uniform approach, which considers minimal resolutions of quotients of gravitational instantons is presented, and they recover a member of nearly all deformation classes of known Ricci-flat hy- perk kähler surfaces with ALF, ALG and ALH asymptotics.
Our project will focus on relating the construction in [BM] to the solutions of the complex Monge-Ampère equation given by [TY1] and [TY2]. On the lines of [S1], which describe the asymptotic behavior of the Tian-Yau metrics, and of [S2], which solves the complex Monge-Ampère equation with prescribed asymptotics, we plan to study conditions for uniqueness of solu- tions to the Monge-Ampère equation, and to use such results to partially classify the remaining Ricci-flat Kähler cases. In particular, we expect to obtain ALF Ricci-flat Kähler metrics on the whole family of deformations of the B
iquard-Minerbe examples, as well as their quotients.