## 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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### References:

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