I was curious about finding conditions for a level surface to be compact and came up with this idea. In particular, let $F(x,y,z): \mathbb{R}^3 \to \mathbb{R}$ be a continuously differentiable function such that $\frac{\partial{F}}{\partial{z}}$ is nowhere vanishing on all of $\mathbb{R}^3$. By the Implicit Function Theorem, the level surface $F = 0$ is then locally the graph of some continuously differentiable function $z = g(x,y)$ about any chosen point in the level surface. If the level surface is moreover the graph of some extension of $g$ on all of $\mathbb{R}^2$, then it cannot be compact, but I am not sure if the condition on $\frac{\partial{F}}{\partial{z}}$ is enough to guarantee this. Does this argument lead anywhere?
One idea I had to find an extension of $g$ is to choose a cover of $\mathbb{R}^2$ and "patch together" the functions given by the implicit function theorem to construct a function on all of $\mathbb{R}^2$.