The geometry of rotation invariant Kähler metrics

The present thesis consists of three results related to the geometry of rotation invariant Kähler metrics. In the first one, we prove that a 3-codimensional Kähler-Einstein submanifold of the complex projective space with rotation invariant metric is forced to be the product of complex projective spaces. In the second one, we prove that the only stable-projectively induced Ricci-flat Kähler metrics are flat. Finally, we prove as third result that given a Ricciflat radial Kähler metric defined on a complex surface such that the third coefficient of its Tian-Yau-Zelditch expansion vanishes, then it is flat.