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A biharmonic maximum principle for hyperbolic surfaces. (English) Zbl 1013.53025

Let \(\Omega\) be a simply connected 2-dimensional Riemannian manifold with a metric \(ds\) represented by \(ds^2(x,y)=\omega(x,y)(dx^2+dy^2)\) for some positive weight function \(\omega\). The manifold \(\Omega\) is said to be hyperbolic if its Gaussian curvature is negative or \(\omega\) is logarithmically subharmonic. A real-valued function \(u\) on \(\Omega\) is biharmonic if \(\Delta ^2u=0\) and sub-biharmonic if \(\Delta ^2u\leq 0\), where \(\Delta\) is the Laplace-Beltrami operator. A maximum principle for \(\Delta ^2\) is \[ \left\{\Delta ^2u_{|D}\leq 0,\;u_{|\partial D}=0,\;\;\text{and} \;\frac{\partial u} {\partial n}_{|\partial D}\leq 0\right\}\Longrightarrow u_{|D}\leq 0, \] where \(D\) is some mean value disk in \(\Omega\) defined by the mean value property \(\int_Dh(z)\Sigma (z)=r^2h(z_0)\) for every bounded harmonic function \(h\) on \(D\), \(d\Sigma \) the area measure on \(\Omega\), \(z_0\in D\) a center, and \(r\) the radius of \(D\).
A conformal map \(f:\mathbb{D}\to D\) pulls the coordinates back to the unit disk, and the geometry of \(\Omega\) then supplies \(\mathbb{D}\) with a hyperbolic geometry. In other words, one can think of \(D\) as being parameterized by the unit disk. Then a simple scaling argument allows to assume that \(\omega\) is reproducing the value \(\int_Dh(z)\omega(z) d\Sigma (z)=h(0)\) at the origin for every bounded harmonic function \(h\) on \(\mathbb{D}\). The biharmonic operator \(\Delta^2\) corresponds to the weighted biharmonic operator \(\Delta\omega^{-1}\Delta\). Let \(\Gamma_\omega\) be the Green function for \(\Delta\omega^{-1}\Delta\) on the unit disk \(\mathbb{D}\). It corresponds to \(\Gamma_D\) by \(f\). In this paper the authors show that if \(\omega\) is a logarithmically subharmonic weight on \(\mathbb{D}\) which reproduces for the origin, then \({\Gamma_\omega}_{|\mathbb{D}\times \mathbb{D}}\geq 0\). The application of this result to Bergman spaces is also presented.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
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