On some aspects of oscillation theory and geometry
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 we prove some new results in both directions, ranging from oscil...
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| Hlavní autoři: | , , |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Providence, Rhode Island
American Mathematical Society
2013
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| Vydání: | 1 |
| Edice: | Memoirs of the American Mathematical Society |
| Témata: | |
| ISBN: | 9780821887998, 0821887998 |
| ISSN: | 0065-9266, 1947-6221 |
| On-line přístup: | Získat plný text |
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- Introduction -- The Geometric setting -- Some geometric examples related to oscillation theory -- On the solutions of the ODE <inline-formula content-type="math/mathml"> ( v z ′ ) ′ + A v z = 0 (vz’)’+Avz=0 </inline-formula> -- Below the critical curve -- Exceeding the critical curve -- Much above the critical curve
- Intro -- Contents -- Chapter 1. Introduction -- Chapter 2. The Geometric setting -- 2.1. Cut-locus and volume growth function -- 2.2. Model manifolds and basic comparisons -- 2.3. Some spectral theory on manifolds -- Chapter 3. Some geometric examples related to oscillation theory -- 3.1. Conjugate points and Myers type compactness results -- 3.2. The spectrum of the Laplacian on complete manifolds -- 3.3. Spectral estimates and immersions -- 3.4. Spectral estimates and nonlinear PDE -- Chapter 4. On the solutions of the ODE ( ')'+ =0 -- 4.1. Existence, uniqueness and the behaviour of zeroes -- 4.2. The critical curve: definition and main estimates -- Chapter 5. Below the critical curve -- 5.1. Positivity and estimates from below -- 5.2. Stability, index of -Δ- ( ) and the uncertainty principle -- 5.3. A comparison at infinity for nonlinear PDE -- 5.4. Yamabe type equations with a sign-changing nonlinearity -- 5.5. Upper bounds for the number of zeroes of -- Chapter 6. Exceeding the critical curve -- 6.1. First zero and oscillation -- 6.2. Comparison with known criteria -- 6.3. Instability and index of -Δ- ( ) -- 6.4. Some remarks on minimal surfaces -- 6.5. Newton operators, unstable hypersurfaces and the Gauss map -- 6.6. Dealing with a possibly negative potential -- 6.7. An extension of Calabi compactness criterion -- Chapter 7. Much above the critical curve -- 7.1. Controlling the oscillation -- 7.2. The growth of the index of -Δ- ( ) -- 7.3. The essential spectrum of -Δ and punctured manifolds -- Bibliography