173 research outputs found
Fractional Sobolev Regularity for the Brouwer Degree
We prove that if is a bounded open set and
, then the Brouwer degree
deg of any H\"older function belongs to the Sobolev space
for every . This extends a
summability result of Olbermann and in fact we get, as a byproduct, a more
elementary proof of it. Moreover we show the optimality of the range of
exponents in the following sense: for every and with
there is a vector field with \mbox{deg}\, (v, \Omega, \cdot)\notin
W^{\beta, p}, where is the unit ball.Comment: 12 pages, 1 figur
Regularity of area minimizing currents II: center manifold
This is the second paper of a series of three on the regularity of higher
codimension area minimizing integral currents. Here we perform the second main
step in the analysis of the singularities, namely the construction of a center
manifold, i.e. an approximate average of the sheets of an almost flat area
minimizing current. Such center manifold is complemented with a Lipschitz
multi-valued map on its normal bundle, which approximates the current with a
highe degree of accuracy. In the third and final paper these objects are used
to conclude a new proof of Almgren's celebrated dimension bound on the singular
set.Comment: In the new version the proofs and the structure are improved and some
minor errors have been correcte
Regularity of area minimizing currents III: blow-up
This is the last of a series of three papers in which we give a new, shorter
proof of a slightly improved version of Almgren's partial regularity of area
minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis
deducing the regularity of area minimizing currents from that of Dir-minimizing
multiple valued functions
The min--max construction of minimal surfaces
In this paper we survey with complete proofs some well--known, but hard to
find, results about constructing closed embedded minimal surfaces in a closed
3-dimensional manifold via min--max arguments. This includes results of J.
Pitts, F. Smith, and L. Simon and F. Smith.Comment: 42 pages, 13 figure
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