Asymptotic Dirichlet Problems for Harmonic Functions on Riemannian Manifolds

DOI10.2307/2000080zbMath0541.53035OpenAlexW4236418737WikidataQ126014045 ScholiaQ126014045MaRDI QIDQ3327289

Hyeong In Choi

Publication date: 1984

Full work available at URL: https://doi.org/10.2307/2000080

zbMATH Keywords

harmonic functionGaussian curvaturecomplete surfacetotal absolute curvatureradial curvatureasymptotic Dirichlet problemmanifold with a poleconvex conic neighbourhood condition

Mathematics Subject Classification ID

Global Riemannian geometry, including pinching (53C20) Harmonic, subharmonic, superharmonic functions on other spaces (31C05) Potential theory on Riemannian manifolds and other spaces (31C12)

Related Items (33)

Dirichlet conditions at infinity for parabolic and elliptic equations ⋮ Asymptotic Dirichlet problem for the $p$-Laplacian on Cartan-Hadamard manifolds ⋮ Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature ⋮ Some observations on Liouville's theorem on surfaces and the Dirichlet problem at infinity ⋮ The Dirichlet problem for the minimal hypersurface equation on arbitrary domains of a Riemannian manifold ⋮ Dirichlet problem at infinity for \(\mathcal A\) -harmonic functions ⋮ Martin compactification of a complete surface with negative curvature ⋮ Complete minimal discs in Hadamard manifolds ⋮ On the asymptotic Dirichlet problem for the minimal hypersurface equation in a Hadamard manifold ⋮ Classification of metric measure spaces and their ends using p-harmonic functions ⋮ AN OBSERVATION ON THE DIRICHLET PROBLEM AT INFINITY IN RIEMANNIAN CONES ⋮ Minimisers of the Allen-Cahn equation and the asymptotic plateau problem on hyperbolic groups ⋮ Brownian motion and Dirichlet problems at infinity ⋮ Convexity at infinity and bounded harmonic functions ⋮ Convexity at infinity in Cartan-Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems ⋮ Dirichlet problem on locally finite graphs ⋮ On the solvability of boundary value problems for the stationary Schrödinger equation in unbounded domains on Riemannian manifolds ⋮ Asymptotic Dirichlet problem for \(\mathcal {A}\)-harmonic functions on manifolds with pinched curvature ⋮ Some results on the convex hull of finitely many convex sets ⋮ Minimisers of the Allen-Cahn equation on hyperbolic graphs ⋮ Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces ⋮ Some existence results on the exterior Dirichlet problem for the minimal hypersurface equation ⋮ Solvability of minimal graph equation under pointwise pinching condition for sectional curvatures ⋮ Brownian motion and the Dirichlet problem at infinity on two-dimensional Cartan-Hadamard manifolds ⋮ Limiting angle of Brownian motion in certain two-dimensional Cartan-Hadamard manifolds ⋮ Asymptotic Dirichlet problem for the Schrödinger operator via rough isometry ⋮ The asymptotic Dirichlet problems on manifolds with unbounded negative curvature ⋮ Geometric and Martin boundaries of a Cartan-Hadamard surface ⋮ Survey on the asymptotic Dirichlet problem for the minimal surface equation ⋮ Rough isometry and the asymptotic Dirichlet problem ⋮ Regularity at infinity of Hadamard manifolds with respect to some elliptic operators and applications to asymptotic Dirichlet problems ⋮ Eigenfunctions of the Laplacian on rotationally symmetric manifolds ⋮ The nonsolvability of the Dirichlet problem on negatively curved manifolds

Cites Work

• Function theory on manifolds which possess a pole
• On subharmonic functions and differential geometry in the large
• The Dirichlet problem at infinity for manifolds of negative curvature
• The Dirichlet problem at infinity for a negatively curved manifold
• Harmonic mappings and minimal submanifolds
• Curvatures of complex submanifolds of \(C^n\)
• On a class of conformal metrics, with application to differential geometry in the large
• Visibility manifolds
• On the Liouville Theorem for Harmonic Maps
• On a new gap phenomenon in riemannian geometry
• Surfaces of nonpositive curvature
• Harmonic functions on complete riemannian manifolds
• On Deciding Whether a Surface is Parabolic or Hyperbolic
• Elliptic Partial Differential Equations of Second Order
• Geodesics and ends in certain surfaces without conjugate points
• Locally Convex Hypersurfaces of Negatively Curved Spaces
• Characterizations of Simply Connected Rotationally Symmetric Manifolds
• Normal families of holomorphic mappings
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