Volume 2014, Issue 686

Abstract. In hyperbolic 3-space surfaces of constant mean curvature H come in three types, corresponding to the cases , , . Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space with and , respectively. These surface classes have been investigated intensively in the literature. For the case there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstrass type representation for surfaces of constant mean curvature in with particular emphasis on the case of mean curvature . In particular, the generalized Weierstrass type representation presented in this paper enables us to construct simultaneously minimal surfaces () and non-minimal constant mean curvature surfaces ().

Abstract. On étudie la partie spécifique de la formule des traces d'Arthur–Selberg pour certains revêtements des groupes réductifs connexes. Comme premier pas vers la formules des traces invariante, on exprime la partie géométrique en termes d'intégrales orbitales pondérées. Les résultats s'appliquent, en particulier, aux revêtements construits par Brylinski et Deligne.

Abstract. Let f be an even weight modular form on a p -adically uniformizable Shimura curve for a suitable -type level structure. Let be an imaginary quadratic field, satisfying Heegner conditions assuring that the sign appearing in the functional equation of the complex L -function of is negative. We may attach to f , or rather a deformation of it, a p -adic L -function of the weight variable κ, also depending on K . Our main result is a formula relating the derivative of this p -adic L -function at to the Abel–Jacobi images of so-called Heegner cycles.

Abstract. We show that the Rost nilpotence principle holds for all surfaces over fields of characteristic zero in the category of Chow motives with integral coefficients.

Abstract. We prove that the generic point of a Hilbert modular four-fold represents an abelian variety which is not a Jacobian. The proof uses degeneration techniques and is independent of properties of the mapping class group used in preceding papers on locally symmetric subvarieties of the moduli space of abelian varieties contained in the Schottky locus.

Abstract. We study the stability of the positive mass theorem using the intrinsic flat distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed intrinsic flat sense. In fact we provide explicit bounds on the intrinsic flat distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the intrinsic flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the intrinsic flat stability of the positive mass theorem in the general case are proposed in the final section.

Abstract. We study real trigonal curves and elliptic surfaces of type  I (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's dessins d'enfants . We give a description of maximally inflected trigonal curves of type  I in terms of the combinatorics of sufficiently simple graphs and, in the case of the rational base, obtain a complete classification of such curves. As a consequence, these results lead to conclusions concerning real Jacobian elliptic surfaces of type  I with all singular fibers real.