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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.
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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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