Morse function

A smooth function with certain special properties. Morse functions arise and are used in Morse theory.

Let $ W $ be a smooth complete (in some Riemannian metric) Hilbert manifold (for example, finite dimensional) whose boundary $ \partial W $ is a disconnected union (possibly empty) of manifolds $ V _ {0} $ and $ V _ {1} $. A Morse function for the triple $ ( W ; V _ {0} , V _ {1} ) $ is a smooth (of Fréchet class $ C ^ {2} $) function $ f : W \rightarrow [ a , b ] $, $ - \infty < a , b < + \infty $( or $ f : W \rightarrow [ a , \infty ] $ for $ V _ {1} = \emptyset $), such that:

1) $ f ^ { - 1 } ( a) = V _ {0} $, $ f ^ { - 1 } ( b) = V _ {1} $;

2) all critical points (cf. Critical point) of $ f $ lie in $ W \setminus \partial W = f ^ { - 1 } ( a, b ) $ and are non-degenerate;

3) condition $ C $ of Palais–Smale is fulfilled (see [2], [3]). I.e. on any closed set $ S \subset W $ where $ f $ is bounded and the greatest lower bound of $ x \rightarrow \| d f ( x) \| $ is zero, there is a critical point of $ f $.

For example, if $ f $ is a proper function, that is, all sets $ f ^ { - 1 } [ c , d ] $, $ - \infty < c , d \leq \infty $, are compact (possible only for $ \mathop{\rm dim} W < \infty $), then $ F $ satisfies condition $ C $. A Morse function attains a (global) minimum on each connected component of $ W $. If $ V $ is a finite-dimensional manifold, then for $ k \geq 2 $ the set of Morse functions of class $ C ^ {k} $ is a set of the second category (and, if $ W $ is compact, even a dense open set) in the space of all functions

$$ f : ( W ; V _ {0} , V _ {1} ) \rightarrow ( [ a , b ] , a , b ) $$

in the $ C ^ {k} $- topology.

References

[1]	M. Morse, "The calculus of variations in the large" , Amer. Math. Soc. (1934)
[2]	R.S. Palais, "Morse theory on Hilbert manifolds" Topology , 2 (1963) pp. 299–340
[3]	S. Smale, "Morse theory and a nonlinear generalization of the Dirichlet problem" Ann. of Math. , 80 (1964) pp. 382–396

Comments

There exist generalizations to Morse functions on stratified spaces (cf. (the editorial comments to) Morse theory and [a1]) and to equivariant Morse functions (cf. [a2] and [a3]).

References

[a1]	M. Goreski, R. MacPherson, "Stratified Morse theory" , Springer (1988)
[a2]	A. Wasserman, "Morse theory for -manifolds" Bull. Amer. Math. Soc. , 71 (1965) pp. 384–388
[a3]	A. Wasserman, "Equivariant differential topology" Topology , 8 (1969) pp. 127–150