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

Abstract

We give a complete list of non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds of a finite dimensional complex projective space endowed with the Fubini-Study metric. This solves in the aforementioned case a classical and long-staying problem addressed among others in [5] and [24].

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

“…Many authors have proved the validity of this conjecture under additional assumptions (see, e.g. [4], [20], [23], [8], [9], [18]). When the ambient space is infinite dimensional the situation changes drastically: there exist continuous families of complete not homogeneous KE metrics projectively induced by an infinite dimensional complex projective space 2 (see [15] and [7]).…”

“…The last assertion in the statement follows immediately by noticing that (18) with G rotation invariant can be satisfied only if the c j 's vanish or n = 1.…”

Fubini-Study geometries in the higher-dimensional gravity

“…In this article, we consider the Fubini-Study metric in four dimensions, which is an exact solutions to the Einstein's equations with a cosmological constant. The Fubini-Study metric has been used to construct the balanced metrics in the context of complex geometry [15], as well as classifying the non-isometric two-dimensional Kahler-Einstein sub-manifolds of a finite dimensional complex projective space [16]. The suggested deep connection between the quantum information theory can be established through the Fubini-Study metric [17,18].…”