Florian Bertrand

DIRICHLET AND NEUMANN PROBLEMS FOR PLANAR DOMAINS WITH PARAMETER

Let ( ·,�) be smooth, i.e. C 1 , embeddings from onto � , where and � are bounded domains with smooth boundary in the complex plane andvaries in I = (0,1). Suppose that is smooth on × I and f is a smooth function on @ × I. Let u(·,�) be the harmonic functions onwith boundary values f(·,�). We show that u(( z,�),�) is smooth on ×I. Our main result is proved for suitable Holder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings ( ·,�) from D, the closure of the unit disc, ontosuch that is smooth on D × I and real analytic at ( √ −1,0) ∈ D × I, but for every family of Riemann mappings R(·,�) fromonto D, the function R(( z,�),�) is not real analytic at ( √ −1,0) ∈ D×I.

Sharp estimates of the Kobayashi metric and Gromov hyperbolicity

Abstract Let D = { ρ 0 } be a smooth relatively compact domain in a four-dimensional almost complex manifold ( M , J ) , where ρ is a J-plurisubharmonic function on a neighborhood of D ¯ and strictly J-plurisubharmonic on a neighborhood of ∂D. We give sharp estimates of the Kobayashi metric. Our approach is based on an asymptotic quantitative description of both the domain D and the almost complex structure J near a boundary point. Following Z.M. Balogh and M. Bonk [Z.M. Balogh, M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000) 504–533], these sharp estimates provide the Gromov hyperbolicity of the domain D.

The star function for meromorphic functions of several complex variables

ABSTRACT We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.

A note on the geometry of pseudoconvex domains of finite type in almost complex manifolds

Let

Almost complex structures on the cotangent bundle

D=\{\rho<0\}

Gromov hyperbolicity of strongly pseudoconvex almost complex manifolds

be a smooth domain of finite type in an almost complex manifold (M,J) of real dimension four. We assume that the defining function

Stationary holomorphic discs and finite jet determination problems

\rho

Riemann-Hilbert problems with constraints

is J-plurisubharmonic on a neighborhood of

Local approximation of non-holomorphic discs in almost complex manifolds

\overline{D}

Gluing techniques and envelopes of disc functionals on almost complex manifolds

. We study the asymptotic behavior of pseudoholomorphic discs contained in the domain D.