3,229 research outputs found
On the Invariant Theory of Weingarten Surfaces in Euclidean Space
We prove that any strongly regular Weingarten surface in Euclidean space
carries locally geometric principal parameters. The basic theorem states that
any strongly regular Weingarten surface is determined up to a motion by its
structural functions and the normal curvature function satisfying a geometric
differential equation. We apply these results to the special Weingarten
surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of
constant Gauss curvature.Comment: 16 page
Quasiconformal Gauss maps and the Bernstein problem for Weingarten multigraphs
We prove that any complete, uniformly elliptic Weingarten surface in
Euclidean -space whose Gauss map image omits an open hemisphere is a
cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman
and Schoen for constant mean curvature surfaces. In particular, this proves
that planes are the only complete, uniformly elliptic Weingarten multigraphs.
We also show that this result holds for a large class of non-uniformly elliptic
Weingarten equations. In particular, this solves in the affirmative the
Bernstein problem for entire graphs for that class of elliptic equations. To
obtain these results, we prove that planes are the only complete multigraphs
with quasiconformal Gauss map and bounded second fundamental form.Comment: 29 pages, 10 figure
On the Integrability of Supersymmetric Versions of the Structural Equations for Conformally Parametrized Surfaces
The paper presents the bosonic and fermionic supersymmetric extensions of the
structural equations describing conformally parametrized surfaces immersed in a
Grasmann superspace, based on the authors' earlier results. A detailed analysis
of the symmetry properties of both the classical and supersymmetric versions of
the Gauss-Weingarten equations is performed. A supersymmetric generalization of
the conjecture establishing the necessary conditions for a system to be
integrable in the sense of soliton theory is formulated and illustrated by the
examples of supersymmetric versions of the sine-Gordon equation and the
Gauss-Codazzi equations
Covariant description of isothermic surfaces
We present a covariant formulation of the Gauss-Weingarten equations and the
Gauss-Mainardi-Codazzi equations for surfaces in 3-dimensional curved spaces.
We derive a coordinate invariant condition on the first and second fundamental
form which is locally necessary and sufficient for the surface to be
isothermic. We show how to construct isothermic coordinates.Comment: 11 page
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