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Register a new userBeurlings free boundary value problem in conformal geometry
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The subject of this paper is Beurlings celebrated extension of the Riemann mapping theorem cite{Beu53}. Our point of departure is the observation that the only known proof of the Beurling-Riemann mapping theorem contains a number of gaps which seem inherent in Beurlings geometric and approximative approach. We provide a complete proof of the Beurling-Riemann mapping theorem by combining Beurlings geometric method with a number of new analytic tools, notably $H^p$-space techniques and methods from the theory of Riemann-Hilbert-Poincare problems. One additional advantage of this approach is that it leads to an extension of the Beurling-Riemann mapping theorem for analytic maps with prescribed branching. Moreover, it allows a complete description of the boundary regularity of solutions in the (generalized) Beurling-Riemann mapping theorem extending earlier results that have been obtained by PDE techniques. We finally consider the question of uniqueness in the extended Beurling-Riemann mapping theorem.
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We prove a version of the Arnold conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.
We study two geometric properties of reproducing kernels in model spaces $K_theta$where $theta$ is an inner function in the disc: overcompleteness and existence of uniformly minimalsystems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern--Clark point. It is shown that uniformly minimal non-Riesz$ $ sequences of reproducing kernelsexist near each Ahern--Clark point which is not an analyticity point for $theta$, whileovercompleteness may occur only near the Ahern--Clark points of infinite orderand is equivalent to a zero localization property. In this context the notion ofquasi-analyticity appears naturally, and as a by-product of our results we give conditions in thespirit of Ahern--Clark for the restriction of a model space to a radius to be a class ofquasi-analyticity.
We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.
For a non-empty compact set $E$ in a proper subdomain $Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $Omega$ by $d(E)$ and $d(E,partialOmega),$ respectively. The quantity $d(E)/d(E,partialOmega)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $Omega$ has the hyperbolic distance $h_Omega(z,w),$ we consider the infimum $kappa(Omega)$ of the quantity $h_Omega(E)/log(1+d(E)/d(E,partialOmega))$ over compact subsets $E$ of $Omega$ with at least two points, where $h_Omega(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $mathbb{H}$. Our main results claim that $kappa(Omega)$ is positive if and only if the boundary of $Omega$ is uniformly perfect and that the inequality $kappa(Omega)leqkappa(mathbb{H})$ holds for all $Omega,$ where equality holds precisely when $Omega$ is convex.
In this paper we establish a connection between free boundary minimal surfaces in a ball in $mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.
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