Free-boundary problems for holomorphic curves in the 6-sphere

Lecture 6: J-holomorphic Curves and Applications

Let (M,ω) be a symplectic manifold of dimension 2n, and let J ∈ J (M,ω) be an ω-compatible almost complex structure. Let gJ(·, ·) ≡ ω(·, J ·) be the corresponding hermitian metric (i.e. J-invariant Riemannian metric) on M . Let (Σ, j) be a Riemann surface (not necessarily compact) with complex structure j. A smooth map u : Σ → M is called a (J, j)-holomorphic map (or simply a J-holomorphic map)...

OSCILLATION AND BOUNDARY CURVATURE OF HOLOMORPHIC CURVES IN Cn

The number of isolated intersections between a smooth curve in Eu-clidean space and an arbitrary hyperplane can be majorized by a weighted sum of integral Frenet curvatures of the curve. In the complex Hermitian space one can derive a similar result for holomorphic curves but with much better weights. The proof of this result is based on a generalizationof the Milnor{FF ary theorem for complex ...

Julia directions for holomorphic curves

A theorem of Picard type is proved for entire holomorphic mappings into projective varieties. This theorem has local nature in the sense that the existence of Julia directions can be proved under natural additional assumptions. An example is given which shows that Borel’s theorem on holomorphic curves omitting hyperplanes has no such local counterpart. Let P be complex projective space of dimen...

On Free Boundary Problems

Free Boundary Problems

Paradigmatic examples are the classical Stefan problem and more general models of phase transitions, where the free boundary is the moving interface between phases. Other examples come from problems in surface science, plastic molding and glass rolling, filtration through porous media, where free boundaries occur as fronts between saturated and unsaturated regions, and others from reaction-diff...