423 research outputs found
Natural PDE's of Linear Fractional Weingarten surfaces in Euclidean Space
We prove that the natural principal parameters on a given Weingarten surface
are also natural principal parameters for the parallel surfaces of the given
one. As a consequence of this result we obtain that the natural PDE of any
Weingarten surface is the natural PDE of its parallel surfaces. We show that
the linear fractional Weingarten surfaces are exactly the surfaces satisfying a
linear relation between their three curvatures. Our main result is
classification of the natural PDE's of Weingarten surfaces with linear relation
between their curvatures.Comment: 16 page
The Codazzi Equation for Surfaces
In this paper we develop an abstract theory for the Codazzi equation on
surfaces, and use it as an analytic tool to derive new global results for
surfaces in the space forms {\bb R}^3, {\bb S}^3 and {\bb H}^3. We give
essentially sharp generalizations of some classical theorems of surface theory
that mainly depend on the Codazzi equation, and we apply them to the study of
Weingarten surfaces in space forms. In particular, we study existence of
holomorphic quadratic differentials, uniqueness of immersed spheres in
geometric problems, height estimates, and the geometry and uniqueness of
complete or properly embedded Weingarten surfaces
On Guichard's nets and Cyclic systems
In the first part, we give a self contained introduction to the theory of
cyclic systems in n-dimensional space which can be considered as immersions
into certain Grassmannians. We show how the (metric) geometries on spaces of
constant curvature arise as subgeometries of Moebius geometry which provides a
slightly new viewpoint. In the second part we characterize Guichard nets which
are given by cyclic systems as being Moebius equivalent to 1-parameter families
of linear Weingarten surfaces. This provides a new method to study families of
parallel Weingarten surfaces in space forms. In particular, analogs of Bonnet's
theorem on parallel constant mean curvature surfaces can be easily obtained in
this setting.Comment: 25 pages, plain Te
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