This thesis aims to discuss some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. In this respect, we prove new results in both directions. For instance, we improve on classical oscillation and nonoscillation criteria for ODE's, and we find sharp spectral estimates for a number of geometric differential operator on Riemannian manifolds. We apply these results to achieve topological and geometric properties. In the first part of the thesis, we collect some material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.

ON SOME ASPECTS OF OSCILLATION THEORY AND GEOMETRY / L. Mari ; relatore: M. Rigoli ; correlatore: B. Bianchini ; coordinatore: M. Peloso. - : . Universita' degli Studi di Milano, 2012 Feb 20. ((24. ciclo, Anno Accademico 2011. [10.13130/mari-luciano_phd2012-02-20].

ON SOME ASPECTS OF OSCILLATION THEORY AND GEOMETRY

L. Mari
2012-02-20

Abstract

This thesis aims to discuss some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. In this respect, we prove new results in both directions. For instance, we improve on classical oscillation and nonoscillation criteria for ODE's, and we find sharp spectral estimates for a number of geometric differential operator on Riemannian manifolds. We apply these results to achieve topological and geometric properties. In the first part of the thesis, we collect some material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.
RIGOLI, MARCO
PELOSO, MARCO MARIA
oscillation ; spectral theory ; index ; Schrodinger operator ; uncertainty principle ; compactness ; immersions ; comparison
Settore MAT/03 - Geometria
Settore MAT/05 - Analisi Matematica
ON SOME ASPECTS OF OSCILLATION THEORY AND GEOMETRY / L. Mari ; relatore: M. Rigoli ; correlatore: B. Bianchini ; coordinatore: M. Peloso. - : . Universita' degli Studi di Milano, 2012 Feb 20. ((24. ciclo, Anno Accademico 2011. [10.13130/mari-luciano_phd2012-02-20].
Doctoral Thesis
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/170621
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