The Martin compactification associated with a
                    second order strictly elliptic partial differential
                    operator on a manifold 𝑀

Taylor, Jasmine

doi:10.1090/crmp/028/06

Cited by 14 publications

(17 citation statements)

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“…For an elliptic operator L, while a Green's function always exists locally, the existence of global Green's function requires extra conditions. The result below follows from [Ta,Corollary 5.13].…”

Section: 2mentioning

confidence: 85%

“…whenever the Green's function G exists, see for example [Ta,Proposition 5.6]. Such a Green's function is called minimal, see [LT] when L = ∆.…”

Section: Preliminariesmentioning

confidence: 99%

“…For any fixed y ∈ M , the function K(·, y) is harmonic on M \ {y} and equals to 1 at x 0 . By the Harnack principle, any sequence Next we collect some basic results on Martin kernels and Martin integral representations, see, e.g., [Mu,Theorem 1.10] and [Ta,Section 6].…”

Section: Preliminariesmentioning

confidence: 99%

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Martin compactification of a complete surface with negative curvature

Cao

2016

Math. Ann.

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Abstract. In this paper we consider the Martin compactification, associated with the operator L = ∆ − 1, of a complete non-compact surface Σ 2 with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator ∆ of Σ 2 and prove a uniqueness result: such eigenfunctions are unique up to a positive constant multiple if they vanish on the part of the geometric boundary S∞(Σ 2 ) of Σ 2 where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of S∞(Σ 2 ) where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper [CH] in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field.

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Section: 2mentioning

confidence: 85%

“…whenever the Green's function G exists, see for example [Ta,Proposition 5.6]. Such a Green's function is called minimal, see [LT] when L = ∆.…”

Section: Preliminariesmentioning

confidence: 99%

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Martin compactification of a complete surface with negative curvature

Cao

2016

Math. Ann.

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“…For perturbation theory, see [Aik1,AM,Anc4,5,Ban1,GH,HZ,M1,3,5,9,10,Pinc1,4,Pins2,Zhao]; as for estimates for fundamental solutions of parabolic equations, see [Aron,Bab,Ban1,2,Ciap,CoSa,Davis,DS,KS,Li,LY,M1,2,7,8,Nor1,2,Pinc2,Sulliv]; and as for general theory for positive solutions of elliptic/parabolic equations, see [Agmon,Anc3,CC,Helms,Phelps,Maeda,Pins2,Sulliv,Taylor].…”

Section: Minoru Murata (Ii) Results On Martin Boundaries and Martin Kmentioning

confidence: 99%

Martin Boundaries of Elliptic Skew Products, Semismall Perturbations, and Fundamental Solutions of Parabolic Equations

Murata

2002

Journal of Functional Analysis

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“…2.7 of [3] or [25]) to show that the process X D has a Martin boundary ∂ M D satisfying the following properties. …”

Section: Martin Kernel and Martin Boundarymentioning

confidence: 99%

Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Kim

Song

2007

Math. Ann.

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The Martin compactification associated with a second order strictly elliptic partial differential operator on a manifold 𝑀

Cited by 14 publications

References 0 publications

Martin compactification of a complete surface with negative curvature

Martin compactification of a complete surface with negative curvature

Martin Boundaries of Elliptic Skew Products, Semismall Perturbations, and Fundamental Solutions of Parabolic Equations

Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

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