On the Dirichlet Problem at Infinity and Poisson Boundary for Certain Manifolds without Conjugate Points

2025/06/28 by Liu, Fei, Zhang, Yinghan
#31C12 #58J32 #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.22883

Abstract

In this paper, we investigate the problem of the existence of the bounded harmonic functions on a simply connected Riemannian manifold \widetildeM without conjugate points, which can be compactified via the ideal boundary \widetildeM(∞). Let \widetildeM be a uniform visibility manifold which satisfy the Axiom 2, or a rank 1 manifold without focal points, suppose that Γ is a cocompact discrete subgroup of Iso(\widetildeM), we show that for a given continuous function on \widetildeM(∞), there exists a harmonic extension to \widetildeM. And furthermore, when \widetildeM is a rank 1 manifold without focal points, the Brownian motion defines a family of harmonic measures ν_∗ on \widetildeM(∞), we show that (\widetildeM(∞),ν_∗) is isomorphic to the Poisson boundary of Γ.

Citations

• On the Myrberg Limit Sets and Bowen-Margulis-Sullivan Measures for Visibility Manifolds without Conjugate Points
• On ergodic properties of geodesic flows on uniform visibility manifolds without conjugate points
• Expansive factors for geodesic flows of compact manifolds without conjugate points and with visibility universal covering
• The Hopf-Tsuji-Sullivan Dichotomy on Visibility Manifolds Without Conjugate Points
• Positive harmonic functions on groups and covering spaces
• On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points
• The Higher Rank Rigidity Theorem for Manifolds With No Focal Points

Related

• Poisson equation on complete manifolds
• Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature
• The Poisson equation on complete manifolds with positive spectrum and applications
• Harmonic functions on rank one asymptotically harmonic manifolds
• Non-Realizability of the Poisson Boundary