open problem - 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 $Sigma_i$, $iin{1,2,3}$)
such that there is no choice of points $p_iinSigma_i$ with pairwise parallel tangent planes.

Let us take a smooth function $f:S^1to mathbb R$, $f(t)approxsin(2cdot t)$ 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. $nabla h_3not=0$ 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 $f:partial Dtomathbb R$; it is easy to check (1).

The graph of $h_3$ is a ruled surface which formed by lines passing
through points $(u,f(u))$, $(phi(u),f(phi(u))inmathbb R^3$, $uin S^1$ for some involution diffeomorphism $phi: S^1to S^1$.
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 :)