A lower bound for first eigenvalue with mixed boundary condition
Let M be an n-dimensional compact Riemannian manifold with or without boundary, and its Ricci curvature Ric M⩾n−1. The paper obtains an inequality for the first eigenvalue η 1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz, Reilly, Escobar and Xia. It is also proved that η 1⩾n for certain n-dimensional compact Riemannian manifolds with boundary, which is an extension of the work of Cheng, Li and Yau.
We consider a family of 4-dimensional Hamiltonian time-periodic linear systems depending on three parameters, λ1, λ2 and e such that for e=0 the system becomes autonomous. Using normal form techniques we study stability and bifurcations for e>0 small enough. We pay special attention to the d'Alembert case. The results are applied to the study of the linear stability of homographic solutions of the planar three-body problem, for some homogeneous potential of degree −α, 0