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