Mean-Value Property on Manifolds With Minimal Horospheres

Todjihounde, Léonard

doi:10.1017/s1446788708000025

Cited by 2 publications

(8 citation statements)

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“…in the studies of the so-called harmonic manifolds and related notions of horospheres and the Lichnerowicz Conjecture. Recall, that a complete Riemannian manifold M is called harmonic if harmonic functions on M satisfy the mean value property, see Willmore [35], Ranjan-Shah [29], also Todjihounde [33] for further definitions and references. Furthermore, see e.g.…”

Section: Harmonic Functionsmentioning

confidence: 99%

“…Let B(x ′′ , ǫ) be a ball in C with x ′′ ∈ γ and x ′′ ∈ B(x, ǫ/2) ∩ B(x 2 , ǫ/2). We apply reasoning at (33) to B(x ′′ , ǫ) using again the mean value property for u 1 and u 2 and obtain that u 1 − u 2 ≡ M in B(x ′′ , ǫ). We continue this procedure along γ till we reach first point x ′′′ ∈ γ, such that…”

Section: The Dirichlet Problem and The Dynamical Programming Principlementioning

confidence: 99%

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Harmonic functions on metric measure spaces

2018

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We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems.Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

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Section: Harmonic Functionsmentioning

confidence: 99%

Section: The Dirichlet Problem and The Dynamical Programming Principlementioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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“…We will use the following stronger version of the maximum principle for the subharmonic functions. Lemma 3.4 of [12] states that: Now we recall the definition of the stability vector field as defined in [31].…”

Section: Strong Liouville Type Propertymentioning

confidence: 99%

“…First we prove the integral formula for the derivative of mean value of a C 1 function on (M, g), a complete, connected, non-compact Riemannian manifold of infinite injectivity radius. The proof uses techniques of the proof of Theorem 3.10 of [31] and proof of Theorem 3.8 of [28].…”

Section: 2mentioning

confidence: 99%

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3-dimensional asymptotically harmonic manifolds with minimal horospheres

Shah

2015

Arch. Math.

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Hemangi Shah would like to dedicate this article to her father, Shri Madhusudan Shah, to ignite a spark of creativity in her.Abstract. (M n , g) be a complete Riemannian manifold without conjugate points. In this paper, we show that if M is also simply connected, then M is flat, provided that M is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of M is shown by using the strongest criterion: {ei} be an orthonormal basis of TpM and {be i } be the corresponding Busemann functions on M ., · · · , be n (x)), is an isometry and therefore, M is flat. Consequently, AH manifolds can have either polynomial or exponential volume growth, generalizing the corresponding result of [23] for harmonic manifolds. In case of harmonic manifold with minimal horospheres (HM), the (second order) flatness was proved in [28] by showing that span{b 2 v |v ∈ TpM } is finite dimensional. We conclude that, the results obtained in this paper are the strongest and wider in comparison to harmonic manifolds, which are known to be AH.In fact, our proof shows the more generalized result, viz.: If (M, g) is a non-compact, complete, connected Riemannian manifold of infinite injectivity radius and of subexponential volume growth, then M is a first order flat manifold.2010 Mathematics Subject Classification. Primary 53C20; Secondary 53C25, 53C35.

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Mean-Value Property on Manifolds With Minimal Horospheres

Cited by 2 publications

References 9 publications

Harmonic functions on metric measure spaces

Harmonic functions on metric measure spaces

3-dimensional asymptotically harmonic manifolds with minimal horospheres

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