Two conjectures on Ricci-flat Kähler metrics

Abstract

We propose two conjectures about Ricci-flat Kähler metrics:Conjecture 1: A Ricci-flat projectively induced metric is flat.Conjecture 2: A Ricci-flat metric on an n-dimensional complex manifold such that the a n+1 coefficient of the TYZ expansion vanishes is flat.We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).2010 Mathematics Subject Classification. 53C55; 58C25; 58F06.

Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms

“…Remark 4. Notice that a Kähler metric which can be Kähler immersed into a non-elliptic (finite or infinite dimensional) complex space form is authomatically c-stable projectivelly induced (the reader is referred to [13] for details).…”

Finite TYCZ expansions and cscK metrics

“…The following proposition, interesting on its own sake, shows that the only radial cscK projectively induced metrics with a 3 = 0 are those just described. It could be interesting to classify all the radial projectively induced cscK metrics without the assumption of the vanishing of a 3 (the reader is referred to [22] for the classification of radial projectively induced Ricci flat Kähler metrics).…”

On the third coefficient of TYZ expansion for radial scalar flat metrics

“…as a function of y, ψ and its derivatives. Therefore, thanks to the formulas (9), (10), (15), (14), (16), (17) and (18), using (5), we convert the PDE…”