Large deviations for Branching Processes in Random Environment

A branching process in random environment $(Z_n, n \in \N)$ is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of {\it large deviations}. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is smaller than the typical geometric growth rate $\bar L$ and $Z_n \ge e^{cn}$ when $c > \bar L$. One way to obtain such an atypical rate of growth is to have a typical realization of the branching process in an atypical sequence of environments. This gives us a general lower bound for the rate of decrease of their probability. When each individual leaves at least one offspring in the next generation almost surely, we compute the exact rate function of these events and we show that conditionally on the large deviation event, the trajectory $t \mapsto \frac1n \log Z_{[nt]}, t\in [0,1]$ converges to a deterministic function $f_c :[0,1] \mapsto \R_+$ in probability in the sense of the uniform norm. The most interesting case is when $c < \bar L$ and we authorize individuals to have only one offspring in the next generation. In this situation, conditionally on $Z_n \le e^{cn}$, the population size stays fixed at 1 until a time $\sim n t_c$. After time $n t_c$ an atypical sequence of environments let $Z_n$ grow with the appropriate rate ($\neq \bar L$) to reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0 on $[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].

The Toledo invariant on smooth varieties of general type

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.Comment: 19 page

Harmonic maps and representations of non-uniform lattices of PU(m,1)

We study representations of lattices of PU(m,1) into PU(n,1). We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m-space to complex hyperbolic n-space. This allows us to give a differential geometric proof of rigidity results obtained by M. Burger and A. Iozzi. We also define a new invariant associated to representations into PU(n,1) of non-uniform lattices in PU(1,1), and more generally of fundamental groups of orientable surfaces of finite topological type and negative Euler characteristic. We prove that this invariant is bounded by a constant depending only on the Euler characteristic of the surface and we give a complete characterization of representations with maximal invariant, thus generalizing the results of D. Toledo for uniform lattices.Comment: v2: the case of lattices of PU(1,1) has been rewritten and is now treated in full generality + other minor modification

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let G be either SU(p,2) with p>=2, Sp(2,R) or SO(p,2) with p>=3. The symmetric spaces associated to these G's are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of K\"{a}hler manifolds and Higgs bundles we study representations of uniform lattices of SU(m,1), m>1, into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G=SU(p,2) with p>=2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p,2)/S(U(p)xU(2)), on which it acts cocompactly

On the equidistribution of totally geodesic submanifolds in compact locally symmetric spaces and application to boundedness results for negative curves and exceptional divisors

We prove an equidistribution result for totally geodesic submanifolds in a compact locally symmetric space. In the case of Hermitian locally symmetric spaces, this gives a convergence theorem for currents of integration along totally geodesic subvarieties. As a corollary, we obtain that on a complex surface which is a compact quotient of the bidisc or of the 2-ball, there is at most a finite number of totally geodesic curves with negative self intersection. More generally, we prove that there are only finitely many exceptional totally geodesic divisors in a compact Hermitian locally symmetric space of the noncompact type of dimension at least 2.Comment: The paper has been substantially rewritten. Corollary 1.3 in the previous versions was false as stated. This has been corrected (see Corollary 1.5). The main results are not affecte

On the second cohomology of K\"ahler groups

Carlson and Toledo conjectured that any infinite fundamental group $\Gamma$ of a compact K\"ahler manifold satisfies $H^2(\Gamma,\R)\not =0$. We assume that $\Gamma$ admits an unbounded reductive rigid linear representation. This representation necessarily comes from a complex variation of Hodge structure (\C-VHS) on the K\"ahler manifold. We prove the conjecture under some assumption on the \C-VHS. We also study some related geometric/topological properties of period domains associated to such \C-VHS.Comment: 21 pages. Exposition improved. Final versio

Roadmap to the morphological instabilities of a stretched twisted ribbon

We address the mechanics of an elastic ribbon subjected to twist and tensile load. Motivated by the classical work of Green and a recent experiment that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon. Different types of instabilities appear in various "corners" of this 4D parameter space, and are addressed through distinct types of asymptotic methods. Our theory employs three instruments, whose concerted implementation is necessary to provide an exhaustive study of the various parameter regimes: (i) a covariant form of the F\"oppl-von K\'arm\'an (cFvK) equations to the helicoidal state - necessary to account for the large deflection of the highly-symmetric helicoidal shape from planarity, and the buckling instability of the ribbon in the transverse direction; (ii) a far from threshold (FT) analysis - which describes a state in which a longitudinally-wrinkled zone expands throughout the ribbon and allows it to retain a helicoidal shape with negligible compression; (iii) finally, we introduce an asymptotic isometry equation that characterizes the energetic competition between various types of states through which a twisted ribbon becomes strainless in the singular limit of zero thickness and no tension.Comment: Submitted to Journal of Elasticity, themed issue on ribbons and M\"obius band