The following problem arises from some trouble I've had regarding a paper by E. Hopf. We have a function $\xi(x,y)$ such that:
- $\xi$ is of class $C^2$ in $\mathbb{R}^2$.
- $\xi_{xx}\xi_{yy} - \xi_{xy}^2 \leq 0$ in $\mathbb{R}^2$.
- $\xi_{xx}\xi_{yy} - \xi_{xy}^2 < 0$ and $\xi_{x} = \xi_y = \xi = 0$ at $(x,y) = (0,0)$.
Note: 2. can be interpreted as saying that the graph of $\xi$ has non-positive Gaussian Curvature everywhere and 3. tells us that at $(0,0)$ it has strictly negative Gaussian Curvature.
My question is if the set $\Omega = \big\{ (x,y):\xi(x,y) \neq 0 \big\}$ can have five (or more) connected components. It is obvious that since the curvature is negative at $(0,0)$ then it must be a saddle point. So, locally, we have four connected components, where $\xi \neq 0$. Then, in the paper, it is easily proven that $\Omega$ cannot have less than 4 connected components. However, Hopf's statement that $\Omega$ has exactly 4 components feels somewhat unsupported.
To me, it feels as if a situation as the one I've drawn below is possible. Where $W$ is a region where $\xi > 0$ surrounded by a (blue) region where $\xi < 0$:
