On some aspects of oscillation theory and geometry

Abstract

Introduction v Chapter 1. The Geometric setting 1.1. Cut-locus and volume growth function 1.2. Model manifolds and basic comparisons 1.3. Some spectral theory on manifolds Chapter 2. Some geometric examples related to oscillation theory 2.1. Conjugate points and Myers type compactness results 2.2. The spectrum of the Laplacian on complete manifolds 2.3. Spectral estimates and immersions 2.4. Spectral estimates and nonlinear PDE Chapter 3. On the solutions of the ODE (vz) + Avz = 0 3.1. Existence, uniqueness and the behaviour of zeroes 3.2. The critical curve: definition and main estimates Chapter 4. Below the critical curve 4.1. Positivity and estimates from below 4.2. Stability, index of −∆ − q(x) and the uncertainty principle 4.3. A comparison at infinity for nonlinear PDE 4.4. Yamabe type equations with a sign-changing nonlinearity 4.5. Upper bounds for the number of zeroes of z Chapter 5. Exceeding the critical curve 5.1. First zero and oscillation 5.2. Comparison with known criteria 5.3. Instability and index of −∆ − q(x) 5.4. Some remarks on minimal surfaces 5.5. Newton operators, unstable hypersurfaces and the Gauss map 5.6. Dealing with a possibly negative potential 5.7. An extension of Calabi compactness criterion Chapter 6. Much above the critical curve 6.1. Controlling the oscillation 6.2. The growth of the index of −∆ − q(x) 6.3. The essential spectrum of −∆ and punctured manifolds Bibliography

On the spectrum of bounded immersions

“…By Proposition 1.21 in [6], h satisfies (H 2 ) whenever the negative part of G is small in the following sense:…”

“…Define f (y) = g(ρ(y)), for some suitable g ∈ C 2 (R + 0 ), g ′ ≥ 0, that will be chosen in a moment. By the Hessian comparison theorem (see [6], Theorem 1.15)…”

A note on Killing fields and CMC hypersurfaces

“…The search for f matching the first requirement of (1.9) can be made via the Laplacian comparison theorem (see for instance [15,20]) by assigning a lower bound on the Ricci tensor of M , and since the behavior at +∞ of a carefully chosen f can be easily detected under this assumption, the second requirement in (1.9) turns out simple to check (we refer the reader to [4] for details). In view of (1.9), we can define the critical curve…”

“…The interested reader is suggested to consult [4] for deepening. Under the assumptions (1.9), we can state the following result: The above results can be extended to constant higher order mean curvatures; we recall that, given the oriented hypersurface F : M m → N m+1 , the Weingarten operator in the direction of the normal ν is the symmetric operator A : T M → T M defined via the identity…”

“…In general L j is not elliptic, but there are geometric sufficient conditions to guarantee this fact; we will be interested in the next two (see Proposition 6.27 in [4])…”