Pathological phenomena in Denjoy-Carleman classes

Let $\mathcal C^M$ denote a Denjoy-Carleman class of $\mathcal C^\infty$ functions (for a given logarithmically-convex sequence $M = (M_n)$). We construct: (1) a function in $\mathcal C^M((-1,1))$ which is nowhere in any smaller class; (2) a function on $\mathbb R$ which is formally $\mathcal C^M$ at every point, but not in $\mathcal C^M(\mathbb R)$; (3) (under the assumption of quasianalyticity) a smooth function on $\mathbb R^p$ ($p \geq 2$) which is $\mathcal C^M$ on every $\mathcal C^M$ curve, but not in $\mathcal C^M(\mathbb R^p)$.Comment: 21 page

Strong illposedness of the incompressible Euler equation in integer CmC^m spaces

We consider the $d$-dimensional incompressible Euler equations. We show strong illposedness of velocity in any $C^m$ spaces whenever $m\ge 1$ is an \emph{integer}. More precisely, we show for a set of initial data dense in the $C^m$ topology, the corresponding solutions lose $C^m$ regularity instantaneously in time. In the $C^1$ case, our proof is based on an anisotropic Lagrangian deformation and a short-time flow expansion. In the $C^m$, $m\ge 2$ case, we introduce a flow decoupling method which allows to tame the nonlinear flow almost as a passive transport. The proofs also cover illposedness in Lipschitz spaces $C^{m-1,1}$ whenever $m\ge 1$ is an integer.Comment: 76 pages. Minor corrections. To appear in GAF

Special Lagrangian m-folds in C^m with symmetries

This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models will be needed to understand Mirror Symmetry and the SYZ conjecture.Comment: 44 pages, LaTeX; (v4) minor corrections and improvement