Is there analogue of four vertex theorem for Riemannian 2-sphere $(S^2,g)$?

Question

When I read the Four Vertex Theorem, I think that there may be analogue for smooth Riemannian 2-sphere $(S^2,g)$. Namely, assume the Gauss curvature of $(S^2,g)$ is $K$, Then, K has at least six extreme points on $(S^2,g)$, is it ?

Answer 1

One way of showing that the Gaussian curvature function on the 2-sphere may have as few as $2$ critical points is to use Gluck's result from

Gluck, Herman. The generalized Minkowski problem in differential geometry in the large. Ann. of Math. (2) 96 (1972), 245–276.

Gluck showed that any positive function on the sphere can be realized as the Gaussian curvature of an embedding in 3-space (in fact his result is more general). Thus, if one chooses a rotationally-symmetric function which is $1$ at the south pole and from there increases to the value $2$ at the north pole (in such a way that the gradient is nonzero everywhere except the two poles), one will obtain a metric with only two critical points (a maximum and a minimum).