3,383 research outputs found
Minimal surfaces in the Heisenberg group
We investigate the minimal surface problem in the three dimensional
Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric.
Using a particular surface measure, we characterize minimal surfaces in terms
of a sub-elliptic partial differential equation and prove an existence result
for the Plateau problem in this setting. Further, we provide a link between our
minimal surfaces and Riemannian constant mean curvature surfaces in H equipped
with different Riemannian metrics approximating the Carnot-Caratheodory metric.
We generate a large library of examples of minimal surfaces and use these to
show that the solution to the Dirichlet problem need not be unique. Moreover,
we show that the minimal surfaces we construct are in fact X-minimal surfaces
in the sense of Garofalo and Nhieu.Comment: 26 pages, 12 figure
A notion of rectifiability modeled on Carnot groups
We introduce a notion of rectifiability modeled on Carnot groups. Precisely,
we say that a subset E of a Carnot group M and N is a subgroup of M, we say E
is N-rectifiable if it is the Lipschitz image of a positive measure subset of
N. First, we discuss the implications of N-rectifiability, where N is a Carnot
group (not merely a subgroup of a Carnot group), which include
N-approximability and the existence of approximate tangent cones isometric to N
almost everywhere in E. Second, we prove that, under a stronger condition
concerning the existence of approximate tangent cones isomorphic to N almost
everywhere in a set E, that E is N-rectifiable. Third, we investigate the
rectifiability properties of level sets of C^1_N functions, where N is a Carnot
group. We show that for almost every real number t and almost every
noncharacteristic point x in a level set of f, there exists a subgroup T_x of H
and r >0 so that f^{-1}(t) intersected with B_H(x,r) is T_x-approximable at x
and an approximate tangent cone isomorphic to T_x at x.Comment: 27 page
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