On the characterization of parabolicity and hyperbolicity of submanifolds

Esteve, A.; Palmer, Vicente

doi:10.1112/jlms/jdr001

Cited by 7 publications

(8 citation statements)

References 20 publications

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“…Our Theorem 5.7 extends to arbitrary weighted manifolds previous criteria of the authors [19] for rotationally symmetric manifolds with weights. In the unweighted case h = 0 we recover previous statements by Esteve and the second author [9], and by Markvorsen and the second author [31]. In Corollary 5.11 we show a weighted version of the hyperbolicity result of Markvorsen and the second author [30] for minimal submanifolds of a Cartan-Hadamard manifold.…”

Section: Introductionsupporting

confidence: 81%

“…Our first result is an extension to the weighted setting of previous theorems for Riemannian manifolds by Esteve and the second author [9], and by Markvorsen and the second author [31]. We note that the particular situation of rotationally symmetric manifolds with weights was analyzed by the authors in Theorems 3.2 and 3.3 of [19].…”

Section: Comparisons Under Bounds On the Sectional Curvaturesmentioning

confidence: 59%

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Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Hurtado

Palmer

Rosales

2020

Journal of Mathematical Analysis and Applications

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Let (M, g) be a complete non-compact Riemannian manifold together with a function e h , which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of M to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in M centered at a point o ∈ M . As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from o. The technique extends to non-compact submanifolds properly immersed in M under certain control on their weighted mean curvature.

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Section: Introductionsupporting

confidence: 81%

Section: Comparisons Under Bounds On the Sectional Curvaturesmentioning

confidence: 59%

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Hurtado

Palmer

Rosales

2020

Journal of Mathematical Analysis and Applications

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“…Now, we are ready to state and prove the main results of this section. The first one is a parabolicity criterion, which in the unweighted case h = 0 follows from a more general statement by Esteve and the second author [26,Thm. 3.4].…”

Section: Parabolicity and Hyperbolicity Results For Submanifoldsmentioning

confidence: 98%

“…By the Cauchy-Schwarz inequality this is guaranteed, for instance, if h is radial and P has bounded h-mean curvature. In the Riemannian context, several parabolicity results for submanifolds have been derived under the hypothesis that the radial mean curvature is bounded; besides the aforementioned references [26] and [47], we refer the reader to [49], [42] and [41].…”

Section: Parabolicity and Hyperbolicity Results For Submanifoldsmentioning

confidence: 99%

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

2020

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Let P be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight e h . The aim of this paper is twofold. First, by assuming certain control on the h-mean curvature of P , we establish comparisons for the h-capacity of extrinsic balls in P , from which we deduce criteria ensuring the h-parabolicity or h-hyperbolicity of P . Second, we employ functions with geometric meaning to describe submanifolds of bounded h-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h-minimal submanifolds appearing in the singularity theory of the mean curvature flow.

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“…The existence of many different conditions implying p−parabolicity is well known and hence, where the p−comparison principle holds (see [12], [11], [18], [4], [8], [13], [17], [10] and also references therein). Among them, there are several works giving conditions in terms of the volume growth of geodesic balls.…”

Section: Introductionmentioning

confidence: 99%

Equivalences among parabolicity, comparison principle and capacity on complete Riemannian manifolds

Aiolfi,

Bonorino,

Ripoll

et al. 2021

Preprint

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In this work we establish new equivalences for the concept of pparabolic Riemannian manifolds. We define a concept of comparison principle for elliptic PDE's on exterior domains of a complete Riemannian manifold M and prove that M is p-parabolic if and only if this comparison principle holds for the p-Laplace equation. We show also that the p-parabolicity of M implies the validity of this principle for more general elliptic PDS's and, in some cases, these results can be extended for non p-parabolic manifolds or unbounded solutions, provided that some growth of these solutions are assumed.

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On the characterization of parabolicity and hyperbolicity of submanifolds

Abstract: Abstract. We give a set of sufficient and necessary conditions for parabolicity and hyperbolicity of a submanifold with controlled mean curvature in a Riemannian manifold with a pole and with sectional curvatures bounded from above or from below.

Cited by 7 publications

References 20 publications

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

Equivalences among parabolicity, comparison principle and capacity on complete Riemannian manifolds

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