Integral of Scalar Curvature on Non-parabolic Manifolds

Xu, Guoyi

doi:10.1007/s12220-019-00174-7

Cited by 10 publications

(8 citation statements)

References 14 publications

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“…We refer to [29,36,38] for some of the recent progress on the problem. Finally, we mention that Theorem 1.4 or its localized version to an end may also be used to show the nonexistence of proper positive Green's functions under a suitable assumption on scalar curvature.…”

Section: Introductionmentioning

confidence: 99%

Geometry of three-dimensional manifolds with scalar curvature lower bound

Munteanu¹,

Wang²

2022

Preprint

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This paper is a continuation of our previous work concerning three-dimensional complete manifolds with scalar curvature bounded from below. One of the purposes is to improve a sharp comparison theorem for the bottom spectrum by removing a volume assumption on unit balls. Another purpose is to derive geometric information when the scalar curvature is assumed to be bounded from below by a positive constant. When the Ricci curvature is asymptotically nonnegative, it is shown that such manifolds must be parabolic and that the lower bound of the scalar curvature is explicitly bounded by the volume of unit balls. In particular, in the case that the Ricci curvature is nonnegative, this implies that the volume of the manifold must grow linearly, which answers a question of Gromov for dimension three.

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Section: Introductionmentioning

confidence: 99%

Geometry of three-dimensional manifolds with scalar curvature lower bound

Munteanu¹,

Wang²

2022

Preprint

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“…A Riemannian manifold M is parabolic if every subharmonic function u on M with u * = sup M u < ∞, must be constant [8,20], equivalently, if every positive superharmonic function u on M is constant. Otherwise M is said to be non-parabolic.…”

Section: Schouten Solitonsmentioning

confidence: 99%

“…, where B r (p) is the open ball with radius r and center at p. For more details see, [20] and also references therein.…”

Section: Schouten Solitonsmentioning

confidence: 99%

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Diameter estimation of gradient $ρ$-Einstein solitons

Shaikh,

Mandal,

Mondal

2021

Preprint

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Our aim in this article is to give a lower bound of the diameter of a compact gradient ρ-Einstein soliton satisfying some given conditions. We have also deduced some conditions of the gradient ρ-Einstein soliton with bounded Ricci curvature to become non-shrinking and non-expanding. Further, we have proved that a complete non-compact gradient shrinking or expanding Schouten soliton with non-constant potential and a boundedness condition on scalar curvature must be non-parabolic.

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“…In fact, Yau proposed a more general version of this problem that is involved with the σ k , k = 1, 2, • • • , n of Ricci tensor in [40]. Unfortunately, Yang [38] constructs a counterexample on Kähler manifold to prove that the general version of Yau's Problem 1.5 does not hold for k = 1, 2, • • • , n − 1, Xu [37] obtains an estimate involved with the integral of scalar curvature towards the Problem 1.5 in the case of three-dimensional Riemannian manifold by using the monotonicity formulas of Colding and Minicozzi [5]. However, Problem 1.5 remains open.…”

Section: Introductionmentioning

confidence: 99%