The number of topologically non-equivalent minimal functions on closed surfaces

Abstract

We consider two classes of smooth functions with three critical values on smooth closed surface of genus $g\geq1$, that possess only one (degenerate) saddle critical point in addition to $k$ local maxima and $l$ local minima. Namely: $C_{k,l}(M_g)$ is the class of such functions on oriented surface $M_g$ and $C_{k,l}(N_g)$ -- on non-oriented surface $N_g$. In this paper we calculate the number of topologically non-equivalent (minimal) functions from the class $C_{1,1}(M_g)$ for all $g\geq 1$
and from the class $C_{1,1}(N_g)$ for $g=5, 6$. Asymptotic estimate for number of topologically non-equivalent functions from the class $C_{1,1}(N_g)$ (as $g\to\infty$) are also established.