A Morse complex for infinite dimensional manifolds - Part I

By Alberto Abbondandolo, Pietro Majer. 2003

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Abstract In this paper and in the forthcoming Part II, we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M , possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle—or more generally an essential subbundle—of the tangent bundle of M , suitably related with the gradient flow of f . This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y : finite dimensionality of W , possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W , classification, up to topological conjugacy, of the gradient flow on the closure of W , in the case dim W = 2 .

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/faa3ae6d19bd8d4a6ec1445aa25ff03c06a0b309