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 $g$.



A generalization of Cauchy's mean value theorem.
The answer is no for $n=2$. It is sufficient to construct 3 surfaces with common boundary (say $\Sigma\_i$, $i\in\{1,2,3\}$)
such that there is no choice of points $p\_i\in\Sigma\_i$ with pairwise parallel tangent planes.


Let us take a smooth function $f:S^1\to \mathbb R$, $f(t)\approx\sin(2\cdot t)$ with one little bump near zero
so it has 3 local minima and maxima.
We want to construct three functions $h\_1,h\_2,h\_3$ from unit disc $D$ to $\mathbb R$ such that
each has $f$ as boundary values and


1. if $\nabla h\_1(x)=\nabla h\_2(y)$ then $\nabla h\_1(x)=0$
2. $\nabla h\_3\not=0$ anywhere in the disc.


Then graphs of functions give the needed surfaces.
The graphs of $h\_1$ and $h\_2$ are parts of boundary of 
convex hull of graph of $f:\partial D\to\mathbb R$; it is easy to check (1).


The graph of $h\_3$ is a ruled surface which formed by lines passing 
through points $(u,f(u))$, $(\phi(u),f(\phi(u))\in\mathbb R^3$, $u\in S^1$ for some involution diffeomorphism $\phi: S^1\to S^1$.
To have the property one has to choose $\phi$ with two fixed points (say at global minima of $f$)
so that if $f(\phi(x))=f(x)$ for some $x$ then $f'(\phi(x)\cdot f'(x)<0$.
The later is easy to arrange, that is the place we need the bump of $f$.


P.S. Hopefully it is correct now :)


