32 research outputs found
Umbilicity and characterization of Pansu spheres in the Heisenberg group
For we define a notion of umbilicity for hypersurfaces in the
Heisenberg group . We classify umbilic hypersurfaces in some cases, and
prove that Pansu spheres are the only umbilic spheres with positive constant
(or horizontal)-mean curvature in up to Heisenberg translations.Comment: 32 pages, 2 figures; in Crelle's journal, 201
Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group
We study immersed, connected, umbilic hypersurfaces in the Heisenberg group
with We show that such a hypersurface, if closed, must
be rotationally invariant up to a Heisenberg translation. Moreover, we prove
that, among others, Pansu spheres are the only such spheres with positive
constant sigma-k curvature up to Heisenberg translations.Comment: 28 pages, 6 figure
A Codazzi-like equation and the singular set for smooth surfaces in the Heisenberg group
In this paper, we study the structure of the singular set for a
smooth surface in the -dimensional Heisenberg group . We
discover a Codazzi-like equation for the -area element along the
characteristic curves on the surface. Information obtained from this ordinary
differential equation helps us to analyze the local configuration of the
singular set and the characteristic curves. In particular, we can estimate the
size and obtain the regularity of the singular set. We understand the global
structure of the singular set through a Hopf-type index theorem. We also
justify that Codazzi-like equation by proving a fundamental theorem for local
surfaces in .Comment: 64 pages, 17 figure
Comparison principles and Liouville theorems for prescribed mean curvature equations in unbounded domains
A Codazzi-like equation and the singular set for  smooth surfaces in the Heisenberg group
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