Volume growth, Green's functions, and parabolicity of ends

DOI10.1215/S0012-7094-99-09714-4zbMath0955.31003MaRDI QIDQ1974904

Ilkka Holopainen

Publication date: 27 March 2000

Published in: Duke Mathematical Journal (Search for Journal in Brave)

zbMATH Keywords

capacityRiemannian manifoldharmonic function

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)

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Cites Work

• On the conformal type of a Riemannian manifold
• A note on \(p\)-subharmonic functions on complete manifolds
• Rough isometries, and combinatorial approximations of geometries of non- compact Riemannian manifolds
• Uniformly elliptic operators on Riemannian manifolds
• Two theorems of N. Wiener for solutions of quasilinear elliptic equations
• On the parabolic kernel of the Schrödinger operator
• Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set
• Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds
• Green's functions on positively curved manifolds
• Harmonic functions and the structure of complete manifolds
• Conformal geometry and quasiregular mappings
• The Wiener test and potential estimates for quasilinear elliptic equations
• Solutions of elliptic equations on manifolds with roughly Euclidean ends
• The Poincaré inequality for vector fields satisfying Hörmander's condition
• The Poisson kernel of positively curved manifolds
• Green's functions, harmonic functions, and volume comparison
• Isometric Riemannian manifolds at infinity
• Local behavior of solutions of quasi-linear equations
• The splitting theorem for manifolds of nonnegative Ricci curvature
• On the theory of covering surfaces.
• THE HEAT EQUATION ON NONCOMPACT RIEMANNIAN MANIFOLDS
• On the existence of a Green's function on a manifold
• Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set
• Positive Solutions of Quasilinear Elliptic Equations on Riemannian Manifolds
• On Complete Manifolds With Nonnegative Ricci Curvature
• Lower curvature bounds, Toponogov's theorem, and bounded topology
• A compactification of a manifold with asymptotically nonnegative curvature
• A note on the isoperimetric constant
• Harmonic functions on complete riemannian manifolds
• Differential equations on riemannian manifolds and their geometric applications
• Harnack inequality and hyperbolicity for subelliptic \(p\)-Laplacians with applications to Picard type theorems
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