Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

Kondo, Kei; Tanaka, Minoru

doi:10.1007/s00208-010-0593-4

Cited by 26 publications

(48 citation statements)

References 27 publications

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“…In fact, the notion of having radial curvature bound has been used by the author in [14,24,25] to investigate some problems like eigenvalue comparisons for the Laplace and p-Laplace operators (between the given complete manifold and its model manifold), the heat kernel comparison, etc. This notion can also be found in other literature (see, for instance, [19,29]). …”

Section: Volume Comparison Theorems For Manifolds With Radial Curvatusupporting

confidence: 71%

The Gagliardo-Nirenberg Inequalities and Manifolds With Non-Negative Weighted Ricci Curvature

Mao

2016

Kyushu J. Math.

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Abstract. In this paper, we show that n-dimensional (n 2) complete and non-compact smooth metric measure spaces with non-negative weighted Ricci curvature in which some Gagliardo-Nirenberg-type inequality holds are not far from the model metric measure n-space (i.e., the Euclidean metric n-space). Moreover, this fact, together with two generalized volume comparison theorems given in [P. Freitas et al. Calc. Var. Partial Differential Equations 51 (2014), 701-724], surprisingly leads to an interesting rigidity theorem for the given metric measure spaces.

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Section: Volume Comparison Theorems For Manifolds With Radial Curvatusupporting

confidence: 71%

The Gagliardo-Nirenberg Inequalities and Manifolds With Non-Negative Weighted Ricci Curvature

Mao

2016

Kyushu J. Math.

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“…Authors [KT1] have recently reached stronger conclusion than Abresch and Gromoll' result above, in which diameter growth condition is replaced by an assumption on total curvatures of model surfaces :…”

Section: Introductionmentioning

confidence: 91%

“…Paraboloids and 2-sheeted hyperboloids are typical examples of a von Mangoldt surface of revolution. An untypical example of a von Mangoldt surface of revolution is found in [KT1,Example 1.2], where its radial curvature function G(γ(t)) changes signs on [0, ∞). We refer to [T1] for other examples of a von Mangoldt surface of revolution.…”

Section: Introductionmentioning

confidence: 99%

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

Kondo

Tanaka

2010

Trans. Amer. Math. Soc.

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Abstract. We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold M with a base point p ∈ M whose radial curvature at p is bounded from below by that of a non-compact model surface of revolution M which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base pointp ∈ M . Notice that our model M does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove the main theorem, we need a new type of the Toponogov comparison theorem. As an application of the main theorem, we present a partial answer to Milnor's open conjecture on the fundamental group of complete open manifolds.

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“…In a series of previous articles (see [KT1], [KT2], and [KT3]), by restricting the total curvature of a noncompact model surface of revolution we investigated some topological properties of a complete and noncompact Riemannian manifold which is not less curved than the model surface. (The precise definition of "not less curved than a noncompact model surface of revolution" is given later in this article.)…”

Section: §1 Introductionmentioning

confidence: 99%

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

Tanaka

Kondo

2013

Nagoya Mathematical Journal

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We will construct peculiar surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we will prove that a complete non-compact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary, if the manifold M is not less curved than a non-compact model surface M of revolution, and if the total curvature of the model surface M is finite and less than 2π.By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete non-compact Riemannian manifolds whose sectional curvatures are bounded from below by a constant. * Mathematics Subject Classification (2010) : 53C21, 53C22. †

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Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

Cited by 26 publications

References 27 publications

The Gagliardo-Nirenberg Inequalities and Manifolds With Non-Negative Weighted Ricci Curvature

The Gagliardo-Nirenberg Inequalities and Manifolds With Non-Negative Weighted Ricci Curvature

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

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