Ball covering property and nonnegative Ricci curvature outside a compact set

Liu, Zhong-dong

doi:10.1090/pspum/054.3/1216638

Cited by 12 publications

(26 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can also ask the same question under other similar hypotheses, for example under the assumption of asymptotically non-negative sectional curvature (see Section 7.5 for the definition). Any of these hypotheses ensures that the manifold has finitely many ends (see [5], [33], [35]). Harmonic functions on such manifolds have been studied, e.g., by Li and Tam [32], [33] and by Kasue [28], but the literature seems to contain no results on (PHI).…”

Section: Tome 55 (2005) Fasciculementioning

confidence: 99%

“…-The manifold M has finitely many ends by [5], [35], [33]. By [35], there exist o ∈ M and a constant Q such that for any r > 0 the set S r = {x ∈ M i : d(o, x) = r} can be covered by at most Q balls of radius Obviously, under (7.34) we have (VC) ⇔ (ψ3). From (ψ3) and (7.35) we obtain It is easy to see that (ψ2) and (ψ3) do not imply each other.…”

Section: Examples Involving Curvature Conditionsmentioning

confidence: 99%

See 1 more Smart Citation

Stability results for Harnack inequalities

Grigor’yan¹,

Saloff‐Coste²

2005

Annales De L’institut Fourier

136

View full text Add to dashboard Cite

Section: Tome 55 (2005) Fasciculementioning

confidence: 99%

Section: Examples Involving Curvature Conditionsmentioning

confidence: 99%

Stability results for Harnack inequalities

Grigor’yan¹,

Saloff‐Coste²

2005

Annales De L’institut Fourier

136

View full text Add to dashboard Cite

“…Thus we have the following results: Note that if a complete Riemannian manifold M with nonnegative Ricci curvature outside a compact subset has finite first Betti number, then M satisfies the conditions (V D) ∞ , (P ) ∞ and (F C) at infinity of each end. (See [18] and [19].) Our result is more general than [17] and [18] in two directions: One is that our assumption contains no curvature assumptions and no topological restrictions like the finiteness of the first Betti number.…”

Section: Theorem 1 Let M Be a Complete Riemannian Manifold With Regulmentioning

confidence: 91%

“…But the condition (F C) holds on a connected sum of complete Riemannian manifolds with nonnegative Ricci curvature. (See [1] and [19]. )…”

Section: L-regularity and Rough Isometric Invariantsmentioning

confidence: 99%

Generalized Liouville property for Schrödinger operator on Riemannian manifolds

Lee¹,

Kim²

2001

Mathematische Zeitschrift

View full text Add to dashboard Cite

In this paper, we prove that the dimension of the space of positive (bounded, respectively) L-harmonic functions on a complete Riemannian manifold with L-regular ends is equal to the number of ends (L-nonparabolic ends, respectively). This result is a solution of an open problem of Grigor'yan related to the Liouville property for the Schrödinger operator L. We also prove that if a given complete Riemannian manifold is roughly isometric to a complete Riemannian manifold satisfying the volume doubling condition, the Poincaré inequality and the finite covering condition on each end, then the dimension of the space of positive (bounded, respectively,) solutions for the Schrödinger operator with a potential satisfying a certain decay rate on the manifold is equal to the number of ends (L-nonparabolic ends, respectively). This is a partial answer of the question, suggested by Li, related to the regularity of ends of a complete Riemannian manifold. Especially, our results directly generalize various earlier results of Yau, of Li and Tam, of Grigor'yan, and of present authors, but with different techniques that the peculiarity of the Schrödinger operator demands.

show abstract

“…Then the volume doubling condition (D) ∞ and the Poincaré inequality (P ) ∞ are satisfied on each end of M. In addition, if the manifold M has finite first Betti number, then the finite covering condition (F C) also holds on each end. (See [20] and [21]. )…”

Section: This Implies Thatmentioning

confidence: 99%

Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds

Lee

2000

Mathematische Annalen

View full text Add to dashboard Cite

We prove that if the s-harmonic boundary of a complete Riemannian manifold consists of finitely many points, then the set of bounded energy finite solutions for certain nonlinear elliptic operators on the manifold is one to one corresponding to R l , where l is the cardinality of the s-harmonic boundary. We also prove that the finiteness of cardinality of s-harmonic boundary is a rough isometric invariant, moreover, in this case, the cardinality is preserved under rough isometries between complete Riemannian manifolds. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Kim and the present author, of Holopainen, and of the present author, but with different techniques which are demanded by the peculiarity of nonlinearity.

show abstract

Ball covering property and nonnegative Ricci curvature outside a compact set

Cited by 12 publications

References 0 publications

Stability results for Harnack inequalities

Stability results for Harnack inequalities

Generalized Liouville property for Schrödinger operator on Riemannian manifolds

Rough isometry and energy finite solutions of elliptic equations on Riemannian manifolds

Contact Info

Product

Resources

About