Morse theory on Banach manifolds

K. Uhlenbeck
1970 Bulletin of the American Mathematical Society  
Let f be a C2 function on a C2 Banach manifold. A critical point x off is said to be weakly nondegenerate if there exists a neighborhood 7J of x and a hyperbolic linear isomorphism L, : T,(M) --, T,(M) such that in the coordinate system of U, df,+,(Lp) > 0 if v # 0. L, defines an index invariantly, and it is shown that this is an extension of the usual definition of nondegeneracy and index. It is shown that this weaker nondegeneracy can be used in place of the stronger nondegeneracy conditions
more » ... n Morse theory. In addition, sufficient conditions for the critical points of variational problems to be weakly nondegenerate are given. Both Morse theory on Hilbert manifolds, developed by Palais and Smale [3, 51, and the Luisternik-Schnirelman theory on Banach manifolds due to Palais [2] are several years old. In an unpublished article [6], S. Smale conjectured that Morse theory can be extended to Banach manifolds. The difficulty in extending the previous methods is in showing the existence of a vector field which is to be transverse to a handle near a critical point. We avoid this problem by assuming the existence of a gradient-like vector field near critical points. This weak nondegeneracy is not a condition which can be easily verified geometrically nor is it necessarily independent of coordinate choice as we define it, but there are natural ways to construct such fields in calculus of variations problems. We are able to obtain a handle-body decomposition of a complete C2 Finsler manifold with C2 partitions of unity which corresponds to a C2 function, if this function satisfies condition C and has weakly nondegenerate critical points. It is also shown that this theory applies to many calculus of variations problems. The question of the genericity of nondegenerate critical points for * An announcement of the results of this paper has appeared as Morse theory on
doi:10.1090/s0002-9904-1970-12384-9 fatcat:qfnianhtcbh4hhymyxc2molfnu