 Three-critical-point function on a torus

Take $u,v$ modulo $\pi$; shifting either coordinate by $\pi$ leaves $h$ unchanged. Its partial derivatives are $h_u=\sin v\sin (2u-v)$ and $h_v=\sin u\sin (u-2v)$. The only critical points are $(0,0)$, $(\pi /3,2\pi /3)$ and $(2\pi /3,\pi /3)$. The latter are a nondegenerate minimum and maximum, with values $\mp 3\sqrt{3}/8$. At the origin the leading cubic is $uv(u-v)$, linearly equivalent to a monkey saddle; its critical point is isolated but degenerate. Thus three smooth critical points coexist with the Morse lower bound of four for the torus.