# On Some Aspects of Oscillation Theory and Geometry

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*Bruno Bianchini; Luciano Mari; Marco Rigoli*

The aim of this paper is to analyze 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. With this motivation the authors prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep their investigation basically self-contained, the authors also collect some, more or less known, material which often appears in the literature in various forms and for which they give, in some instances, new proofs according to their specific point of view.

#### Table of Contents

# Table of Contents

## On Some Aspects of Oscillation Theory and Geometry

- Chapter 1. Introduction 18 free
- Chapter 2. The Geometric setting 714 free
- Chapter 3. Some geometric examples related to oscillation theory 4148
- Chapter 4. On the solutions of the ODE (𝑣𝑧’)’+𝐴𝑣𝑧=0 6976
- Chapter 5. Below the critical curve 8794
- Chapter 6. Exceeding the critical curve 121128
- 6.1. First zero and oscillation 121128
- 6.2. Comparison with known criteria 127134
- 6.3. Instability and index of -Δ-𝑞(𝑥) 130137
- 6.4. Some remarks on minimal surfaces 132139
- 6.5. Newton operators, unstable hypersurfaces and the Gauss map 140147
- 6.6. Dealing with a possibly negative potential 151158
- 6.7. An extension of Calabi compactness criterion 153160

- Chapter 7. Much above the critical curve 163170
- Bibliography 187194