Sharp uncertainty principles on Riemannian manifolds: The influence of curvature

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Abstract

We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:. (a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn. Since the sharp HPW principle and the Hardy-Poincare´ inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds. Nous pre´sentons un sce´nario de rigidite´ pour les varie´te´s riemanniennes complètes soutenant le principe d'incertitude d'Heisenberg-Pauli-Weyl avec la constante optimale en Rn (brièvement, le principle d'HPW). Nos re´sultats de´pendent profonde´ment de la courbure de la varie´te´ riemannienne et ils peuvent eˆtre formule´s comme suit :. (a)Lorsque (M,g) a courbure sectionnelle non positive, le principe d'HPW a lieu sur (M,g). Ne´anmoins, des fonctions extre´males positives existent dans le principe d'HPW si et seulement si (M,g) est isome´trique à Rn, n=dim(M).(b)Lorsque (M,g) a courbure de Ricci non ne´gative, le principe d'HPW a lieu sur (M,g) si et seulement si (M,g) est isome´trique à Rn. Comme le principe d'HPW et l'ine´galite´ Hardy-Poincare´ sont des cas extreˆmes de l'ine´galite´ d'interpolation de Caffarelli-Kohn-Nirenberg, nous e´tablissons des re´sultats quantitatifs pour les dernières ine´galite´s en terme de la courbure sur les varie´te´s de Cartan-Hadamard.

Original languageEnglish
JournalJournal des Mathematiques Pures et Appliquees
DOIs
Publication statusAccepted/In press - Jun 7 2017

Fingerprint

Uncertainty Principle
Isometric
Riemannian Manifold
Interpolation
Caffarelli-Kohn-Nirenberg Inequalities
Curvature
Interpolation Inequality
Hadamard Manifolds
If and only if
Sharp Constants
Nonpositive Curvature
Scenarios
Hardy Inequality
Nonnegative Curvature
Poincaré Inequality
Ricci Curvature
Sectional Curvature
Rigidity
Poincaré
Extremes

Keywords

  • Curvature
  • Heisenberg-Pauli-Weyl uncertainty principle
  • Primary
  • Riemannian manifold
  • Secondary
  • Sharp constant

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

@article{f5176d5a7b464db08d68eff36fa8b37a,
title = "Sharp uncertainty principles on Riemannian manifolds: The influence of curvature",
abstract = "We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:. (a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn. Since the sharp HPW principle and the Hardy-Poincare´ inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds. Nous pre´sentons un sce´nario de rigidite´ pour les varie´te´s riemanniennes compl{\`e}tes soutenant le principe d'incertitude d'Heisenberg-Pauli-Weyl avec la constante optimale en Rn (bri{\`e}vement, le principle d'HPW). Nos re´sultats de´pendent profonde´ment de la courbure de la varie´te´ riemannienne et ils peuvent eˆtre formule´s comme suit :. (a)Lorsque (M,g) a courbure sectionnelle non positive, le principe d'HPW a lieu sur (M,g). Ne´anmoins, des fonctions extre´males positives existent dans le principe d'HPW si et seulement si (M,g) est isome´trique {\`a} Rn, n=dim(M).(b)Lorsque (M,g) a courbure de Ricci non ne´gative, le principe d'HPW a lieu sur (M,g) si et seulement si (M,g) est isome´trique {\`a} Rn. Comme le principe d'HPW et l'ine´galite´ Hardy-Poincare´ sont des cas extreˆmes de l'ine´galite´ d'interpolation de Caffarelli-Kohn-Nirenberg, nous e´tablissons des re´sultats quantitatifs pour les derni{\`e}res ine´galite´s en terme de la courbure sur les varie´te´s de Cartan-Hadamard.",
keywords = "Curvature, Heisenberg-Pauli-Weyl uncertainty principle, Primary, Riemannian manifold, Secondary, Sharp constant",
author = "A. Krist{\'a}ly",
year = "2017",
month = "6",
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language = "English",
journal = "Journal des Mathematiques Pures et Appliquees",
issn = "0021-7824",
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TY - JOUR

T1 - Sharp uncertainty principles on Riemannian manifolds

T2 - The influence of curvature

AU - Kristály, A.

PY - 2017/6/7

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N2 - We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:. (a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn. Since the sharp HPW principle and the Hardy-Poincare´ inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds. Nous pre´sentons un sce´nario de rigidite´ pour les varie´te´s riemanniennes complètes soutenant le principe d'incertitude d'Heisenberg-Pauli-Weyl avec la constante optimale en Rn (brièvement, le principle d'HPW). Nos re´sultats de´pendent profonde´ment de la courbure de la varie´te´ riemannienne et ils peuvent eˆtre formule´s comme suit :. (a)Lorsque (M,g) a courbure sectionnelle non positive, le principe d'HPW a lieu sur (M,g). Ne´anmoins, des fonctions extre´males positives existent dans le principe d'HPW si et seulement si (M,g) est isome´trique à Rn, n=dim(M).(b)Lorsque (M,g) a courbure de Ricci non ne´gative, le principe d'HPW a lieu sur (M,g) si et seulement si (M,g) est isome´trique à Rn. Comme le principe d'HPW et l'ine´galite´ Hardy-Poincare´ sont des cas extreˆmes de l'ine´galite´ d'interpolation de Caffarelli-Kohn-Nirenberg, nous e´tablissons des re´sultats quantitatifs pour les dernières ine´galite´s en terme de la courbure sur les varie´te´s de Cartan-Hadamard.

AB - We present a rigidity scenario for complete Riemannian manifolds supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn (shortly, sharp HPW principle). Our results deeply depend on the curvature of the Riemannian manifold which can be roughly formulated as follows:. (a)When (M,g) has non-positive sectional curvature, the sharp HPW principle holds on (M,g). However, positive extremals exist in the sharp HPW principle if and only if (M,g) is isometric to Rn, n=dim(M).(b)When (M,g) has non-negative Ricci curvature, the sharp HPW principle holds on (M,g) if and only if (M,g) is isometric to Rn. Since the sharp HPW principle and the Hardy-Poincare´ inequality are endpoints of the Caffarelli-Kohn-Nirenberg interpolation inequality, we establish further quantitative results for the latter inequalities in terms of the curvature on Cartan-Hadamard manifolds. Nous pre´sentons un sce´nario de rigidite´ pour les varie´te´s riemanniennes complètes soutenant le principe d'incertitude d'Heisenberg-Pauli-Weyl avec la constante optimale en Rn (brièvement, le principle d'HPW). Nos re´sultats de´pendent profonde´ment de la courbure de la varie´te´ riemannienne et ils peuvent eˆtre formule´s comme suit :. (a)Lorsque (M,g) a courbure sectionnelle non positive, le principe d'HPW a lieu sur (M,g). Ne´anmoins, des fonctions extre´males positives existent dans le principe d'HPW si et seulement si (M,g) est isome´trique à Rn, n=dim(M).(b)Lorsque (M,g) a courbure de Ricci non ne´gative, le principe d'HPW a lieu sur (M,g) si et seulement si (M,g) est isome´trique à Rn. Comme le principe d'HPW et l'ine´galite´ Hardy-Poincare´ sont des cas extreˆmes de l'ine´galite´ d'interpolation de Caffarelli-Kohn-Nirenberg, nous e´tablissons des re´sultats quantitatifs pour les dernières ine´galite´s en terme de la courbure sur les varie´te´s de Cartan-Hadamard.

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KW - Primary

KW - Riemannian manifold

KW - Secondary

KW - Sharp constant

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