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.In an attempt to give a unified approach to a number of apparently different geometric problems, based on the notion of critical curve, the paper is organized as follows. ... on the behaviour of the function vol(aBr) that shall determine the regularity of the coefficients in the Cauchy problem (CP). ... The chapter ends with a short review of spectral theory on manifolds. ... Jorge and Montenegro [BJM10], which positively answers a question of S.T. Yau on the discreteness of the spectrum ofanbsp;...
|Title||:||On Some Aspects of Oscillation Theory and Geometry|
|Author||:||Bruno Bianchini, Luciano Mari, Marco Rigoli|
|Publisher||:||American Mathematical Soc. - 2013-08-23|