4,514 research outputs found
Equality in Borell-Brascamp-Lieb inequalities on curved spaces
By using optimal mass transportation and a quantitative H\"older inequality,
we provide estimates for the Borell-Brascamp-Lieb deficit on complete
Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb
inequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities)
are characterized in terms of the optimal transport map between suitable
marginal probability measures. These results provide several qualitative
applications both in the flat and non-flat frameworks. In particular, by using
Caffarelli's regularity result for the Monge-Amp\`ere equation, we {give a new
proof} of Dubuc's characterization of the equality in Borell-Brascamp-Lieb
inequalities in the Euclidean setting. When the -dimensional Riemannian
manifold has Ricci curvature for some , it turns out that equality in the Borell-Brascamp-Lieb inequality is
expected only when a particular region of the manifold between the marginal
supports has constant sectional curvature . A precise characterization is
provided for the equality in the Lott-Sturm-Villani-type distorted
Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not
necessarily reversible) Finsler manifolds are also presented.Comment: 28 pages (with 1 figure); to appear in Advances in Mathematic
Weak contact equations for mappings into Heisenberg groups
Let k>n be positive integers. We consider mappings from a subset of
k-dimensional Euclidean space R^k to the Heisenberg group H^n with a variety of
metric properties, each of which imply that the mapping in question satisfies
some weak form of the contact equation arising from the sub-Riemannian
structure of the Heisenberg group. We illustrate a new geometric technique that
shows directly how the weak contact equation greatly restricts the behavior of
the mappings. In particular, we provide a new and elementary proof of the fact
that the Heisenberg group H^n is purely k-unrectifiable. We also prove that for
an open set U in R^k, the rank of the weak derivative of a weakly contact
mapping in the Sobolev space W^{1,1}_{loc}(U;R^{2n+1}) is bounded by almost
everywhere, answering a question of Magnani. Finally we prove that if a mapping
from U to H^n is s-H\"older continuous, s>1/2, and locally Lipschitz when
considered as a mapping into R^{2n+1}, then the mapping cannot be injective.
This result is related to a conjecture of Gromov.Comment: 28 page
Geometric inequalities on Heisenberg groups
We establish geometric inequalities in the sub-Riemannian setting of the
Heisenberg group . Our results include a natural sub-Riemannian
version of the celebrated curvature-dimension condition of Lott-Villani and
Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin
to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\"ager. The
latter statement implies sub-Riemannian versions of the geodesic
Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on
optimal mass transportation and Riemannian approximation of
developed by Ambrosio and Rigot. These results refute a general point of view,
according to which no geometric inequalities can be derived by optimal mass
transportation on singular spaces.Comment: to appear in Calculus of Variations and Partial Differential
Equations (42 pages, 1 figure
Star formation activity of intermediate redshift cluster galaxies out to the infall regions
We present a spectroscopic analysis of two galaxy clusters out to ~4Mpc at
z~0.2. The two clusters VMF73 and VMF74 identified by Vikhlinin et al. (1998)
were observed with MOSCA at the Calar Alto 3.5m telescope. Both clusters lie in
the ROSAT PSPC field R285 and were selected from the X-ray Dark Cluster Survey
(Gilbank et al. 2004) that provides optical V- and I-band data. VMF73 and VMF74
are located at respective redshifts of z=0.25 and z=0.18 with velocity
dispersions of 671 km/s and 442 km/s, respectively. The spectroscopic
observations reach out to ~2.5 virial radii. Line strength measurements of the
emission lines H_alpha and [OII]3727 are used to assess the star formation
activity of cluster galaxies which show radial and density dependences. The
mean and median of both line strength distributions as well as the fraction of
star forming galaxies increase with increasing clustercentric distance and
decreasing local galaxy density. Except for two galaxies with strong H_alpha
and [OII] emission, all of the cluster galaxies are normal star forming or
passive galaxies. Our results are consistent with other studies that show the
truncation in star formation occurs far from the cluster centre.Comment: 15 pages, 12 figures. A&A in pres
Quasiconformal mappings that highly distort dimensions of many parallel lines
We construct a quasiconformal mapping of -dimensional Euclidean space, , that simultaneously distorts the Hausdorff dimension of a nearly
maximal collection of parallel lines by a given amount. This answers a question
of Balogh, Monti, and Tyson.Comment: 12 page
Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group
We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean C2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in H is provided
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