Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic K?hler manifold, we define the symplectic Wirtinger number W(X). We show that W(X) \leqslant 1,W(X) \leqslant 1, and the equality is reached if and only if the subvariety X ì MX \subset M is trianalytic, i.e. compatible with the hyperk?hler structure on M. For a sequence X₁ ? X₂ ? ?X_n ? MX_1 \to X_2 \to \ldots X_n \to M of immersions of simple holomorphic symplectic manifolds, we show that W( X₁ ) \leqslant W( X₂ ) \leqslant ?\leqslant W( X_n ).W\left( {X_1 } \right) \leqslant W\left( {X_2 } \right) \leqslant \ldots \leqslant W\left( {X_n } \right).     

On the Mattila Integral Associated with Sign Indefinite Measures

In order to quantitatively illustrate the rôle of positivity in the Falconer distance problem, we construct a family of sign indefinite, compactly supported measures in \({\Bbb R}^d\), such that their Fourier transform and Fourier energy of dimension \(s \in (0, d)\) are uniformly bounded. However, the Mattila integral, associated with the Falconer distance problem for these measures is unbounded in the range \(0 < s < \frac{d^2}{2d-1}\).     

Dynamo theory,vorticity generation,and exponential stretching

A discussion is given of the analogy between the dynamo equation for the generation of a magnetic field by the motion of an electrically conducting fluid and the equation for the evolution of vorticity of a viscous fluid. In both cases exponential stretching is an important feature of the underlying instability problem. For the "fast" dynamo problem, the existence of exponential stretching (i.e., the positivity of the Lyapunov exponent) somewhere in the flow is a necessary condition when the flow is smooth. An example is presented of a flow with exponential stretching (an Anosov flow) that supports fast dynamo action. A parallel treatment is described for the linearized Navier-Stokes equations for the motion of a viscous fluid. In this problem the analogous necessary condition for "fast vorticity generation" is the existence of some instability in the corresponding Euler (i.e., inviscid) equation. Dynamo theory methods give a second related result, namely a universal geometric estimate from below on the growth rate of a small perturbation in an inviscid fluid. This bound gives an effective sufficient condition for local instability for Eulers equations. In particular, it is proved that a steady flow with a hyperbolic stagnation point is unstable. The growth rate of an infinitesimal perturbation in a metric with derivatives depends on this metric. This dependence is completely described.     

Trace Inequalities of Sobolev Type in the Upper Triangle Case

We give a new non-capacitary characterization of positive Borelmeasures µ on Rⁿ such that the potential space I*L^p isimbedded in L^q(dµ) for $1qp+, that is, the trace inequality holds, for Riesz potentials I = (- )². A weak-type trace inequality is also characterized. A non-isotropic version on the unit sphere Sⁿ is studied,as well as the holomorphic case for Hardy–Sobolev spaces in the ball. 1991 MathematicsSubject Classification: primary 31C15, 42B20; secondary 32A35.     

Existence and regularity of positive solutions of elliptic equations of Schr?dinger type

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? ⁿ under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ^??(??) ^n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.     

Trianalytic Subvarieties of the Hilbert Scheme of Points on a K3 Surface

Let X be a hyperk?hler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperk?hler structure. Given a K3 surface M, the Hilbert scheme classifying zero-dimensional subschemes of M admits a hyperk?hler structure. We show that for M generic, there are no trianalytic subvarieties of the Hilbert scheme. This implies that a generic deformation of the Hilbert scheme of K3 has no proper complex subvarieties. Submitted: May 1997, Revised version: December 1998     

Plateau-Stein manifolds

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all ?∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.     

Incompressible Flows of an Ideal Fluid¶with Unbounded Vorticity

In this paper we study solutions to the Euler equations of an ideal incompressible fluid in R ⁿ singular at the origin with a finite symmetry group. For an “admissible” class of finite groups we prove a local existence and uniqueness theorem. In even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity. In arbitrary dimension (including n=3) we construct local in time solutions with vorticity that behaves, e.g., like a function of homogeneous degree zero near the origin. The symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness. Received: 31 March 1999 / Accepted: 10 July 2000