Liz Vivas

Geodesics in the space of Kähler metrics

Let (X,ω) be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set H0 of Kähler forms cohomologous to ω has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H0 can be connected by a smooth geodesic, and show that the answer, in general, is “no”.

ATTRACTING BASINS OF VOLUME PRESERVING AUTOMORPHISMS OF ℂk

We study topological properties of attracting sets for automorphisms of ℂk. Our main result is that a generic volume preserving automorphism has a hyperbolic fixed point with a dense stable manifold. On the other hand, we show that an attracting set can only contain a neighborhood of the fixed point if it is an attracting fixed point. We will see that the latter does not hold in the non-autonomous setting.

Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics

Using Green’s hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi–Huang and Baouendi–Ebenfelt–Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric

Fatou-Bieberbach Domains as Basins of Attraction of Automorphisms Tangent to the Identity

Q(A,B)

Dynamics of two-resonant biholomorphisms

Q(A,B), either the image of the mapping is contained in a complex affine subspace, or

Parametrization of unstable manifolds for parabolic skew-products

A