The following simple theorem is known as Cauchy's mean value theorem. Let $\gamma$ be an immersion of the segment $[0,1]$ into the plane such that $\gamma(0) \ne \gamma(1)$. Then there exists a point such that the tangent line at that point is parallel to the line passing through $\gamma(0)$ and $\gamma(1)$. So the boundary values of an immersion determine a direction such that for any immersion of a segment with given boundary values there exists a tangent line parallel to the direction.

I would like to propose the following conjecture, generalizing this statement. Instead of immersions of a segment we consider immersions of a compact manifold $M^n$ with non-empty boundary $\partial M$ into ${\mathbf R}^{n+1}$. For a map $f\colon \partial M \to {\mathbf R}^{n+1}$ we consider the space $L(f)$ of all immersions $g\colon M \to {\mathbf R}^{n+1}$ such that $g|_{\partial M}=f$.

The conjectural claim is the following: If $f$ is sufficiently generic, then for every connected component $L_0$ of the space $L(f)$ there exists a hyperplane direction $l$ such that for any immersion $g$ from $L_0$ $l$ is parallel to the tangent plane to $g(M)$ at some point.

If the conjecture is true then it is very interesting how $l$ depends on $g$.

`$M=[0,1)$`

looks like a counterexample to me. What reasons do you have for the conjecture to be true? The next simplest case after the intervals would be the disk, I think. Have you tried proving your conjecture for that case? (I am not a differential geometer, so I can't contribute to a solution. But I am curious all the same.) – Harald Hanche-Olsen Feb 25 '10 at 2:19