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Existence of common zeros for commuting vector fields on -manifolds
In E. Lima proved that commuting vector fields on surfaces with non-zero
Euler characteristic have common zeros. Such statement is empty in dimension
, since all the Euler characteristics vanish. Nevertheless,
\cite{Bonatti_analiticos} proposed a local version, replacing the Euler
characteristic by the Poincar\'e-Hopf index of a vector field in a region
, denoted by ; he asked:
\emph{Given commuting vector fields and a region where
, does contain a common zero of and
?}
\cite{Bonatti_analiticos} gave a positive answer in the case where and
are real analytic.
In this paper, we prove the existence of common zeros for commuting
vector fields , on a -manifold, in any region such that
, assuming that the set of collinearity of
and is contained in a smooth surface. This is a strong indication that the
results in \cite{Bonatti_analiticos} should hold for -vector fields.Comment: Final version, to appear in Annales de L'Institut Fourie
Variations along the Fuchsian locus
The main result is an explicit expression for the Pressure Metric on the
Hitchin component of surface group representations into PSL(n,R) along the
Fuchsian locus. The expression is in terms of a parametrization of the tangent
space by holomorphic differentials, and it gives a precise relationship with
the Petersson pairing. Along the way, variational formulas are established that
generalize results from classical Teichmueller theory, such as Gardiner's
formula, the relationship between length functions and Fenchel-Nielsen
deformations, and variations of cross ratios.Comment: 58 pages, 1 figur
A novel type of Sobolev-Poincar\'e inequality for submanifolds of Euclidean space
For functions on generalised connected surfaces (of any dimensions) with
boundary and mean curvature, we establish an oscillation estimate in which the
mean curvature enters in a novel way. As application we prove an a priori
estimate of the geodesic diameter of compact connected smooth immersions in
terms of their boundary data and mean curvature. These results are developed in
the framework of varifolds. For this purpose, we establish that the notion of
indecomposability is the appropriate substitute for connectedness and that it
has a strong regularising effect; we thus obtain a new natural class of
varifolds to study. Finally, our development leads to a variety of questions
that are of substance both in the smooth and the nonsmooth setting.Comment: 35 pages, no figure
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