On Some Aspects of Oscillation Theory and Geometry.

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.

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