We prove the existence of extremal, nonconstant-scalar curvature, Kahler metrics on certain unstable projectivized vector bundles P(E) -> M over a compact constant scalar curvature Kahler manifold M with discrete holomorphic automorphism group, in certain adiabatic Kahler classes. In particular, the vector bundles E -> M are assumed to split as a direct sum of stable subbundles E = El circle plus ... circle plus E-s all having different Mumford-Takemoto slope, for example, mu(E-1) > ... > mu(E-s).

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.

We establish a long exact sequence for Legendrian submanifolds L⊂P×R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H* maps to linearized contact cohomology CH*, which maps to linearized contact homology CH*, which maps to singular homology. In particular, the sequence implies a duality between Ker(CH*→H*) and CH*/Im(H*). Furthermore, this duality is compatible with Poincaré duality in L in the following sense: the Poincaré dual of a singular class which is the image of a∈CH* maps to a class α∈CH* such that α(a)=1.

The exact sequence generalizes the duality for Legendrian knots in R3 (see [26]) and leads to a refinement of the Arnold conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7]

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.

Johnson, T.

Chalmers, Fraunhofer Chalmers Res Ctr Ind Math, S-41296 Gothenburg, Sweden..

Martens, M.

SUNY Stony Brook, Dept Math, Stony Brook, NY 11794 USA..

The period-doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated, as was shown previously. The Jacobian rigidity conjecture says that the period-doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, for example, the one-dimensional case. The other extreme case is when the maps preserve area, for example, when the average Jacobian is one. Indeed, the main result presented here is that the period-doubling Cantor sets of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Let G = SL(2, R) proportional to R-2 be the affine special linear group of the plane, and set Gamma = SL(2, Z) proportional to Z(2). We prove a polynomially effective asymptotic equidistribution result for the orbits of a 1-dimensional, nonhorospherical unipotent flow on Gamma\G.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

We study the value distribution of the Epstein zeta function E-n (L, s) for 0 <s < n/2 and a random lattice L of large dimension n. For any fixed c is an element of (1/4, 1/2) and n -> infinity, we prove that the random variable V-n(-2c) E-n(.,cn) has a limit distribution, which we give explicitly (here V-n is the volume of the n-dimensional unit ball). More generally, for any fixed epsilon > 0, we determine the limit distribution of the random function c bar right arrow V-n(-2c) E-n(., cn), c epsilon [1/4 + epsilon, 1/2 - epsilon]. After compensating for the pole at c = 1/2, we even obtain a limit result on the whole interval [1/4 + epsilon, 1/2], and as a special case we deduce the following strengthening of a result by Sarnak and Strombergsson concerning the height function h(n) (L) of the flat torus R-n/L: the random variable n{h(n) (L) - (log(4 pi) - gamma + 1)} + log n has a limit distribution as n -> infinity, which we give explicitly. Finally, we discuss a question posed by Sarnak and Strombergsson as to whether there exists a lattice L subset of R-n for which E-n(L, s) has no zeros in (0, infinity).