2. Teichmüller’s theorem: If X is compact, or more generally if X is obtained from a compact Riemann surface by removing a finite number of points, then there is an essentially map f minimizing the infimum above. (This map is unique up to conformal automorphisms of X and Y which are isotopic to the identity). The unique map is characterized by the fact that its Beltrami coefficient μ = ∂f/∂f is...

The Teichmüller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case th...

A central method in the theory of quadratic forms is the study of function fields of projective quadrics. In particular, it is important to ask when there is a rational map from one quadric over a field to another. This suggests the problem of determining when two quadrics are birational, which turns out to be much harder. The answer is known for quadratic forms of dimension at most 7 (thus for...

We will investigate the stability behavior of quadratic maps in higher dimensions. To check stability, we will use infinitesimal V -stability of critical points of the map; since the infinitesimal V -stability of a map at all of its critical points is equivalent to the stability of the map. We will establish the connection between infinitesimal V-stability of solutions to the Algebraic Riccati ...

We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Hénon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the ...