We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov exponent. In accordance with the famous Sullivan dictionary, this leads to a description of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.

It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya’s conjecture. Together with a previous work by Avila, this gives the sharp description of phase transitions for the AMO for the a.e. phase.

We establish sharp trace Sobolev inequalities of order four on Euclidean $d$-balls for $d\ge 4$. When $d=4$, our inequality generalizes the classical second-order Lebedev–Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we characterize the extremal metric of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls, which surprisingly is not the flat metric on the ball.

In 1982, Demailly showed that the Hodge conjecture follows from the statement that all positive closed currents with rational cohomology class can be approximated by positive linear combinations of integration currents. Moreover, in 2012, he showed that the Hodge conjecture is equivalent to the statement that any $(p,p)$-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents. In this article, we find a current which does not verify the former statement on a smooth projective variety for which the Hodge conjecture is known to hold. To construct this current, we extend the framework of “tropical currents”—recently introduced by the first author—from tori to toric varieties. We discuss extremality properties of tropical currents and show that the cohomology class of a tropical current is the recession of its underlying tropical variety. The counterexample is obtained from a tropical surface in ${\mathbb{R}}^4$ whose intersection form does not have the right signature in terms of the Hodge index theorem.