Suppose a continuously differentiable function $f$ on a Riemannian manifold $M$, $f:M\rightarrow \mathbb{R}$, has a set of local minimum points $S$ with $f(x)=f(y)$, for all $x,y \in S$. Is it possible to conclude that there exists a level set $\{x \in M\,|\,f(x)\leq h\}$ for some $h\in\mathbb{R}$ which has a connected component $C$ such that $C\cap S\neq\emptyset$ and $f(z)\geq f(x)$ for all $z\in C$?
Context: I would like to show that a set of minimizers of $f$ are stable with respect to the gradient descent flow of $f$. I know that the flow will remain in connected components of the level set for all future times.