304,863 research outputs found
Holomorphic flexibility properties of compact complex surfaces
We introduce the notion of a stratified Oka manifold and prove that such a
manifold is strongly dominable in the sense that for every , there
is a holomorphic map f:\C^n\to X, , such that and is a
local biholomorphism at 0. We deduce that every Kummer surface is strongly
dominable. We determine which minimal compact complex surfaces of class VII are
Oka, assuming the global spherical shell conjecture. We deduce that the Oka
property and several weaker holomorphic flexibility properties are in general
not closed in families of compact complex manifolds. Finally, we consider the
behaviour of the Oka property under blowing up and blowing down.Comment: Version 2: Theorem 11 reformulated and its proof corrected. Minor
improvements to the exposition. Version 3: A few minor improvements. To
appear in International Mathematics Research Notice
The singular set of mean curvature flow with generic singularities
A mean curvature flow starting from a closed embedded hypersurface in
must develop singularities. We show that if the flow has only generic
singularities, then the space-time singular set is contained in finitely many
compact embedded -dimensional Lipschitz submanifolds plus a set of
dimension at most . If the initial hypersurface is mean convex, then all
singularities are generic and the results apply.
In and , we show that for almost all times the evolving
hypersurface is completely smooth and any connected component of the singular
set is entirely contained in a time-slice. For or -convex hypersurfaces
in all dimensions, the same arguments lead to the same conclusion: the flow is
completely smooth at almost all times and connected components of the singular
set are contained in time-slices. A key technical point is a strong
{\emph{parabolic}} Reifenberg property that we show in all dimensions and for
all flows with only generic singularities. We also show that the entire flow
clears out very rapidly after a generic singularity.
These results are essentially optimal
Zero Sets of Solutions to Semilinear Elliptic Systems of First Order
Consider a nontrivial solution to a semilinear elliptic system of first order
with smooth coefficients defined over an -dimensional manifold. Assume the
operator has the strong unique continuation property. We show that the zero set
of the solution is contained in a countable union of smooth -dimensional
submanifolds. Hence it is countably -rectifiable and its Hausdorff
dimension is at most . Moreover, it has locally finite -dimensional
Hausdorff measure. We show by example that every real number between 0 and
actually occurs as the Hausdorff dimension (for a suitable choice of
operator). We also derive results for scalar elliptic equations of second
order.Comment: 16 pages, LaTeX2e, 2 figs, uses pstricks macro packag
Dynamical surface structures in multi-particle-correlated surface growths
We investigate the scaling properties of the interface fluctuation width for
the -mer and -particle-correlated deposition-evaporation models. These
models are constrained with a global conservation law that the particle number
at each height is conserved modulo . In equilibrium, the stationary
roughness is anomalous but universal with roughness exponent ,
while the early time evolution shows nonuniversal behavior with growth exponent
varying with models and . Nonequilibrium surfaces display diverse
growing/stationary behavior. The -mer model shows a faceted structure, while
the -particle-correlated model a macroscopically grooved structure.Comment: 16 pages, 10 figures, revte
On the rationality of quadric surface bundles
For any standard quadric surface bundle over , we show that the
locus of rational fibres is dense in the moduli space.Comment: 20 pages; to appear in Annales de l'Institut Fourie
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