In the talk we discuss some results of [1].
We apply our previous methods [2]
to geometry and to the mappings with bounded distortion.
\textbf{Theorem 1}. Let $v:\Omega\to\mathbb{R}$ be a
$C^1$-smooth function on a domain (open connected set)
$\Omega\subset\mathbb{R}^2$. Suppose
$$ (1)\qquad \operatorname{Int} \nabla v(\Omega)=\emptyset. $$
Then $\operatorname{meas}\nabla v(\Omega)=0$.
Here $\operatorname{Int}E$ is the interior of ${E}$,
$\operatorname{meas} E$ is the Lebesgue measure of ${E}$.
Theorem 1 is a straight consequence of the following two results.
\textbf{ Theorem 2 [2]}. Let $v:\Omega\to\mathbb{R}$ be a
$C^1$-smooth function on a domain $\Omega\subset\mathbb{R}^2$.
Suppose (1) is fulfilled. Then the graph of $v$ is a normal
developing surface.
Recall that a $C^1$-smooth manifold $S\subset\mathbb{R}^3$ is
called a normal developing surface [3] if for any
$x_0\in S$ there exists a straight segment $I\subset S$ (the point
$x_0$ is an interior point of $I$) such that the tangent plane to
$S$ is stationary along $I$.
\textbf{Theorem 3}. The spherical image of any
$C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ has
the area (the Lebesgue measure) zero.
Recall that the spherical image of a surface $S$ is the
set $\{\mathbf{n}(x)\mid x\in S\}$, where $\mathbf{n}(x)$ is the
unit normal vector to $S$ at the point~$x$.
From Theorems 1--3 and the classical results of A.V. Pogorelov
(see [4, Chapter 9]) we obtain the following corollaries Corollary 4. Let the spherical image of a
$C^1$-smooth surface $S\subset\mathbb{R}^3$ have no interior
points. Then this surface is a surface of zero extrinsic curvature
in the sense of Pogorelov.
\textbf{ Corollary 5}. Any $C^1$-smooth normal developing
surface $S\subset\mathbb{R}^3$ is a surface of zero extrinsic
curvature in the sense of Pogorelov.
\textbf{Theorem 6}. Let
$K\subset\mathbb{R}^{2\times
2}$ be a compact set and the topological dimension
of $K$ equals 1. Suppose there exists $\lambda> 0$ such that $\forall A,B\in K, \, \,
|A-B|^2\le\lambda\det(A-B).$
Then for any Lipschitz mapping $f:\Omega\to\mathbb R^2$
on a domain $\Omega\subset\mathbb R^2$ such that $\nabla f(x)\in K$ a.e.
the identity
$\nabla f\equiv\operatorname{const}$
holds.
Many partial cases of Theorem 6 (for instance, when $K=SO(2)$ or $K$
is a segment) are well-known (see, for example, [5]).
The author was supported by the Russian Foundation for
Basic Research (project no. 08-01-00531-a).
[1] {Korobkov M.\,V.,} {``Properties of the $C^1$-smooth functions
whose gradient range has topological dimension~1,'' Dokl. Math.,
to appear.}
[2] {Korobkov M.\,V.} {``Properties of the $C^1$-smooth functions
with nowhere dense gradient range,'' Siberian Math. J.,
\textbf{48,} No.~6, 1019--1028 (2007).}
[3] { Shefel${}'$ S.\,Z.,} {``$C^1$-Smooth isometric
imbeddings,'' Siberian Math. J., \textbf{15,} No.~6, 972--987
(1974).}
[4] {Pogorelov A.\,V.,} {Extrinsic geometry of convex surfaces,
Translations of Mathematical Monographs. Vol. 35. Providence,
R.I.: American Mathematical Society (AMS). VI (1973).}
[5] {M\"uller ~S.,} {Variational Models for Microstructure and
Phase Transitions. Max-Planck-Institute for Mathematics in the
Sciences. Leipzig (1998) (Lecture Notes, No.~2.
http://www.mis.mpg.de/jump/publications.html).}
