Kähler–Einstein submanifolds of the infinite dimensional projective space

Abstract

This paper consists of two main results. In the first one we describe all Kähler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and nonhomogeneous Kähler-Einstein metric with negative scalar curvature which admits a Kähler immersion into the infinite dimensional complex projective space.

Some remarks on the Kähler geometry of the Taub-NUT metrics

“…Consider for instance a Cartan domain of genus γ endowed with its Bergman metric g B . Then αg B is balanced if and only if α > (γ − 1)/γ (see [35,37] for a proof).…”

“…Then, it does exist also a Kähler immersion into CP N of the Kähler submanifold of (C 2 , ω m ) defined by z 2 = 0, z 1 = z, endowed with the induced metric, having potential˜ m = u 2 + mu 4 , where u is defined implicitly by zz = e 2mu 2 u 2 . Observe that˜ m is the Calabi's diastasis function for this metric, since it is a rotation invariant potential centered at the origin (see [11] or also [35,Th. 3,p…”

On holomorphic isometric immersions of nonhomogeneous Kähler–Einstein manifolds into the infinite dimensional complex projective space

“…The bounded symmetric domain (D 1 ×D 2 , c 1 g 1 ⊕c 2 g 2 ) in Theorem 2.4 is a Hermitian symmetric space of noncompact type. The problems of immersing the irreducible Hermitian symmetric spaces (which contain compact type, noncompact type and flat type) into complex space forms have been completely resolved (see [17,18,25]). Those results are also summarized in the end of [25].…”

“…Therefore, it is natural to ask: Does there exist a complete nonhomogeneous Kähler-Einstein submanifold in CP ∞ ? In fact, Loi and Zedda [18] have proved that there do exist such noncompact submanifolds of CP ∞ , namely, Cartan-Hartogs domains (see [35] for the definition of Cartan-Hartogs domain). More precisely, they proved that on the Cartan-Hartogs domains, the explicit Kähler-Einstein metrics obtained by Yin, Roos, the second and the third authors in [35] and [34] are exact projectively induced by using Calabi's criterion (see Theorem 4.2 for the details).…”

Kähler immersions of homogeneous Kähler manifolds into complex space forms

“…(M, g) such that ω is an integral form but g is not projectively induced. In order to describe such an example we recall the following result (see Theorem 2 in [LZ11]). …”