The vanishing of the log term of the Szego kernel and Tian–Yau–Zelditch expansion

This thesis consists in two results. In [Z. Lu, G. Tian, The log term of Szego kernel, Duke Math. J. 125, N 2 (2004), 351-387], the authors conjectured that given a Kähler form ω on CPn in the same cohomology class of the Fubini–Study form ωFS and considering the hyperplane bundle (L; h) with Ric(h) = ω, if the log–term of the Szego kernel of the unit disk bundle Dh L vanishes, then there is an automorphism φ : CPn → CPn such that φω = ωFS. The first result of this thesis consists in showing a particular family of rotation invariant forms on CP2 that confirms this conjecture. In the second part of this thesis we find explicitly the Szego kernel of the Cartan–Hartogs domain and we show that this non–compact manifold has vanishing log–term. This result confirms the conjecture of Z. Lu for which if the coefficients aj of the TYZ expansion of the Kempf distortion function of a n– dimensional non–compact manifold M vanish for j > n, then the log–term of the disk bundle associated to M is zero

Helix surfaces for Berger-like metrics on the anti-de Sitter space

We consider the Anti-de Sitter space $\mathbb{H}^3_1$ equipped with Berger-like metrics, that deform the standard metric of $\mathbb{H}^3_1$ in the direction of the hyperbolic Hopf vector field. Helix surfaces are the ones forming a constant angle with such vector field. After proving that these surfaces have (any) constant Gaussian curvature, we achieve their explicit local description in terms of a one-parameter family of isometries of the space and some suitable curves. These curves turn out to be general helices, which meet at a constant angle the fibers of the hyperbolic Hopf fibration

On the Szegö kernel of Cartan–Hartogs domains

Inspired by the work of Z. Lu and G. Tian (Duke Math. J. 125:351–387, 2004) in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan–Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan–Hartogs domain can be viewed as an “iterated” disk bundle over its base and the ideas given in (Arezzo, Loi and Zuddas in Math. Z. 275:1207–1216, 2013) for the computation of the Szegö kernel of the disk bundle over an Hermitian symmetric space of compact type