Morse theory for Min-type functions

Abstract

0. Introduction. Morse theory for distance functions was initiated by GroveShiohama [GS] and Gromov’s [G1] paper where basic notions of the theory were formulated; even this very initial level of the theory leads to important geometric applications. Grove and Shiohama [GS] established a generalized sphere theorem by constructing a vector field on a Riemannian manifold, with the property that the distance function had no stationary points along the integral curves at non-singular points of the vector field. They showed that this vector field had exactly two singular points and hence the manifold was homeomorphic to a sphere. Later Gromov [G1] was able to bound the sum of the Betti numbers of a positively curved Riemannian manifold. He controlled the location of critical points of a Riemannian distance function on a positively curved manifold using Toponogov’s theorem and then was able to bound the number of critical points of this function and hence the homology of the manifold, using a spectral sequence argument. Morse theory for Riemannian distance functions was discussed in [Gr], [L] and other papers, which explained its importance for geometric applications rather than developed the theory itself. Even a suitable concept of the index of a critical point has not been developed. As a result, the most powerful tools of the classical Morse theory such as Morse inequalities and the correspondence between critical points and the handle decomposition of the manifold cannot be used. The relationship with the classical Morse theory has not been investigated either; in particular, the connection between notions of the critical points in both theories is not obvious. In a different direction, the structure of Alexandrov spaces with curvatures bounded below was investigated using distance functions in several papers starting with [BGP] and continuing with [P1], [P2]. A key result obtained is a canonical stratification of such Alexandrov spaces into topological manifolds and again the technique is a type of Morse theory, using the distance function. As Alexandrov spaces do not have as much structure as Riemannian manifolds, our theory gives more detailed information on the nature of critical points and index. M. Gromov pointed out in [G1] that the Morse theory for Riemannian distance functions can be developed by analogy with the classical Morse theory. The aim of this paper is to construct a Morse theory for functions which are minima of finite families of smooth functions and clarify the connection with Riemannian distance functions for non-positively curved manifolds. We develop the theory in the classical style, including the notion of the index for critical points, and clarify relations with the Grove-Shiohama-Gromov approach.