a Morse function whose total number of critical points is less or than equal to that of any other Morse function on M?
r]-norm, then [phi] is also a Morse function with the same number of critical points as h.
Moreover, if it is Morse it is a minimal Morse function.
On the other hand, it is a general fact that the sum of the betti numbers of a manifold M is a lower bound for the number of critical points of any Morse function on M.
Each nonzero linear form in these variables is a Morse function with two critical points, which is the minimal possible, since this is precisely the sum of the betti numbers of [S.
In this article a novel manifestation of the role played by the heat equation in mathematics is pointed out; namely, that in some homogeneous riemannian manifolds, evolution by the heat equation of a generic initial datum "discovers all by itself" a minimal Morse function for the manifold, i.
A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres
Specifically, we present two concrete cases in which the Heat Equation "discovers all by itself" minimal Morse functions.
Writing to be accessible to first-year graduate students specializing in topology and geometry, Pajitnov covers Morse functions
and vector fields on manifolds, transversality, handles, Morse complexes, cellular gradients, circle-valued more maps and Novikov complexes.