This is the first of three articles designed to stabilize the global trace formula. The results apply to any group for which the fundamental lemma (and its variants for weighted orbital integrals) is valid. The main purpose of this paper is to establish a series of expansions that are parallel to the expansions in the trace formula. We shall also formulate the local and global theorems required to interpret the terms in these expansions. The proofs of the theorems will be given in the subsequent two articles. The expansions of this paper will then yield both a stable trace formula, and a decomposition of the ordinary trace formula into a linear combination of stable trace formulae.

AMS 2000 Mathematics subject classification: Primary 22E55. Secondary 11R39

A necessary and sufficient geometric condition is established for the Kohn Laplacian to be hypoelliptic, modulo its nullspace, for boundaries of arbitrary infinitely differentiable pseudoconvex tube domains in two complex variables. Hypoellipticity is also proved to be equivalent to validity of a superlogarithmic estimate, for this special class of structures.

AMS 2000 Mathematics subject classification: Primary 32W05; 32T27. Secondary 35J70.

We prove that given a convex Jordan curve $\varGamma\subset\{x_3=0\}$, the space of properly embedded minimal annuli in the half-space $\{x_3\geq0\}$, with boundary $\varGamma$ is diffeomorphic to the interval $[0,\infty)$. Moreover, for a fixed positive number $a$, the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper half-space, with finite total curvature, boundary $\varGamma$ and a catenoid type end with logarithmic growth $a$ has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.

AMS 2000 Mathematics subject classification: Primary 53A10. Secondary 49Q05; 53C42

Nori’s connectivity theorem compares the cohomology of $X\times B$ and $Y_B$, where $Y_B$ is any locally complete quasiprojective family of sufficiently ample complete intersections in $X$. When $X$ is the projective space, and we consider hypersurfaces of degree $d$, it is possible to give an explicit bound for $d$, sufficient to conclude that the Connectivity Theorem holds. We show that this bound is optimal, by constructing for lower $d$ classes on $Y_B$ not coming from the ambient space. As a byproduct we get the non-triviality of the higher Chow groups of generic hypersurfaces of degree $2n$ in $\mathbb{P}^{n+1}$.

AMS 2000 Mathematics subject classification: Primary 14C25; 14D07; 14J70