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

Abstract

We exhibit families of non trivial (i.e. not Kähler-Einstein) radial Kähler-Ricci solitons (KRS), both complete and not complete, which can be Kähler immersed into infinite dimensional complex space forms. This result shows that the triviality of a KRS induced by a finite dimensional complex space form proved in [12] does not hold when the ambient space is allowed to be infinite dimensional. Moreover, we show that the radial potential of a radial KRS induced by a non-elliptic complex space form is necessarily defined at the origin. ContentsAppendix A. The proofs of Lemma 2.3 and Lemma 2.4 19 References 21

On holomorphic isometries into blow-ups of $\mathbb C^n$

“…Proof of (b) The proof follows the same line of the proof of (iv) in Theorem 1.1 by using in (16) the function (17) instead of ϕ(z) − q 2 .…”

“…Regarding (ii) observe that, despite the fact that a KE submanifold of a non elliptic complex space form is forced to be totally geodesic [25], the classification of the KE submanifolds of the complex projective space is still missing (the reader is referred to [19,Chapter 3] for an updated material on this subject). In [18] (see also [16] and [17]) the authors have conjectured that these manifolds are indeed open subset of flag manifolds (i.e. compact and simply-connected homogeneous Kähler manifolds).…”

On canonical radial Kaehler metrics

“…Introduced by Calabi [1], are those metrics such that the (1,0)-part of the Hamiltonian vector field associated to the scalar curvature is holomorphic. The reader is referred to [10] and references therein for more details. We denote by Ext(M ) the set of extremal metrics on M .…”

2-dimensional Kähler-Einstein metrics induced by finite dimensional complex projective spaces

“…Our only assumption involves the group of symmetries of the metric. Indeed, our goal will be to test the above mentioned conjecture in the case of rotation invariant Kähler metrics (see also [13] for a list of projectively induced extremal metrics in the radial 4 case). Definition 2.…”

“…3 The classification of Kähler-Einstein manifolds admitting an immersion into an infinite dimensional complex space form is an open problem in all three cases (for some partial results see e.g. [6,13,14,17]). 4 I.e.…”