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Pathological phenomena in Denjoy-Carleman classes
Let denote a Denjoy-Carleman class of
functions (for a given logarithmically-convex sequence ). We
construct: (1) a function in which is nowhere in any
smaller class; (2) a function on which is formally
at every point, but not in ; (3) (under the assumption
of quasianalyticity) a smooth function on () which is
on every curve, but not in .Comment: 21 page
Strong illposedness of the incompressible Euler equation in integer spaces
We consider the -dimensional incompressible Euler equations. We show
strong illposedness of velocity in any spaces whenever is an
\emph{integer}. More precisely, we show for a set of initial data dense in the
topology, the corresponding solutions lose regularity
instantaneously in time. In the case, our proof is based on an
anisotropic Lagrangian deformation and a short-time flow expansion. In the
, 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 whenever 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
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