Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds


The central result of this paper is a generalization of the Ehresmann fibration theorem to the infinite-dimensional and/or non-proper setting. With this aim, we introduce the concepts of strong submersion and of mapping with uniformly split kernels. First applications include a global Implicit Function Theorem and a necessary and sufficient version of Hadamard’s global invertibility criterion in the setting of Finsler manifolds. Another major application makes use of the idea of asymptotic critical value (not critical point), which helps formulate various generalizations of the Palais-Smale condition for morphisms of Finsler manifolds, not merely functionals. We obtain a critical point theory for morphisms of Finsler manifolds extending many results known only for functionals, notably Ekeland’s “variational” principle. These results are used to discuss a new approach to Lagrange multiplier and nonlinear eigenvalue problems, and to develop an intrinsic critical point theory for complex-analytic functionals. The latter reveals an intimate connection between conditions of Palais-Smale type and the structure of polynomial automorphisms (Jacobian Conjecture).


Patrick J. Rabier