Harmonic functions and the structure of complete manifolds

Abstract. We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and nonexplosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties, etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds, etc., which are related to recurrence and non-explosion.

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“…See [40], [50], [95], [96], [119], [121], [123], [124], [175] for further results on harmonic functions on manifolds with ends.…”

Section: (D)mentioning

confidence: 99%

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

Grigor’yan

1999

Bull. Amer. Math. Soc.

699

734

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“…More precisely, Anderson deduces from (i) of Theorem 1 that M is planar provided it has only one end and this is the case because of condition (ii). Indeed, the potential theoretic characterization of the ends by H.-D. Cao, Y. Shen and S. Zhu, [11], shows that all the ends of M are non-parabolic so that, by harmonic function theory, [17], [18], condition (ii) of Theorem 1 forces M to have only one end; see also [22]. Unlike Anderson, Shen-Zhu show that (i) of Theorem 1 forces a uniform decay of |II| 2 that is faster than expected, i.e., sup M \B R |II| 2 = O (R −m ).…”

Section: Analysis Of the Proofs Of The Geometric Theoremmentioning

confidence: 99%

REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Pigola

Veronelli

2012

Int. J. Math.

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Abstract. We survey some L p -vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schrödinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces of finite total curvature is observed. Further geometric applications are discussed. Introduction and some vanishing resultsThis paper originates from an attempt to understand the abstract content of a structure theorem for stable minimal hypersurfaces of finite total scalar curvature.(ii) The "stability operator" L = −∆ − |II| 2 has non-negative spectrum.Then, |II| ≡ 0, that is, f (M ) is an affine m-plane .

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“…By the theory of [8], {f R } converges (by passing to a subsequence if necessary) to a nonconstant harmonic function f with finite Dirichlet integral on M as R → +∞.…”

Section: Manifolds With Positive Spectrummentioning

confidence: 99%

Spectrum of the Laplacian on manifolds with Spin(9) holonomy

Lam

2008

Mathematical Research Letters

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Abstract. We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result. In the second part, we established a splitting type theorem by utilizing the Busemann function.

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Harmonic functions and the structure of complete manifolds

Cited by 181 publications

References 26 publications

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Spectrum of the Laplacian on manifolds with Spin(9) holonomy

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Harmonic functions and the structure of complete manifolds

Cited by 181 publications

References 26 publications

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

REMARKS ON Lp-VANISHING RESULTS IN GEOMETRIC ANALYSIS

Spectrum of the Laplacian on manifolds with Spin(9) holonomy

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REMARKS ON L^p-VANISHING RESULTS IN GEOMETRIC ANALYSIS