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Bounded harmonic functions on Riemannian manifolds of nonpositive curvature. (English) Zbl 1262.53034
Certain general conditions (I–IV) are imposed on a complete simply-connected Riemannian manifold \(M\) of nonpositive curvature, which guarantee that \(M\) supports nontrivial bounded harmonic functions.
Very roughly speaking the conditions (I–IV) are as follows. The sectional curvature of \(M\) is pinched between a negative constant and zero, the Ricci curvature of \(M\) is less than a negative constant. Removing a certain point \(O\in M\), there exists a topological fibration \(M \setminus \{ O\} = \mathcal{A} \times \mathcal {B}\), such that \(\mathcal{N}=\mathcal{A} \times [0, \infty) \) is a complete submanifold of \(M\) and \(O = (a^O, 0)\) for some \(a^O \in \mathcal{A}\). Some geometrical properties of the projection of \(M \setminus \{O\}\) to \(\mathcal{N}\) (i.e. suitable relations between the angle and its segments, for geodesic triangles) and some pinching on the sectional curvature of \(\mathcal{N}\) are required.
The main result provides new families of bounded harmonic functions on \(M\). In particular, it includes the Cartan–Hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains \(\operatorname{Re}_I(n,n)\) and \(\operatorname{Re}_{II}(n)\;(n\geqslant 2)\) as special cases.
MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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