2,886 research outputs found
Limit of Fractional Power Sobolev Inequalities
We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the
4-sphere as the limiting cases of the fractional power Sobolev inequalities on
the same spaces, and justify our approach as the dimensional continuation
argument initiated by Thomas P. Branson.Comment: 17 page
Some higher order isoperimetric inequalities via the method of optimal transport
In this paper, we establish some sharp inequalities between the volume and
the integral of the -th mean curvature for -convex domains in the
Euclidean space. The results generalize the classical Alexandrov-Fenchel
inequalities for convex domains. Our proof utilizes the method of optimal
transportation.Comment: 21 page
On fractional GJMS operators
We describe a new interpretation of the fractional GJMS operators as
generalized Dirichlet-to-Neumann operators associated to weighted GJMS
operators on naturally associated smooth metric measure spaces. This gives a
geometric interpretation of the Caffarelli--Silvestre extension for
when , and both a geometric interpretation
and a curved analogue of the higher order extension found by R. Yang for
when . We give three applications of this
correspondence. First, we exhibit some energy identities for the fractional
GJMS operators in terms of energies in the compactified Poincar\'e--Einstein
manifold, including an interpretation as a renormalized energy. Second, for
, we show that if the scalar curvature and the fractional
-curvature of the boundary are nonnegative, then the
fractional GJMS operator is nonnegative. Third, by assuming
additionally that is not identically zero, we show that
satisfies a strong maximum principle.Comment: 38 pages. Final version, to appear in Communications on Pure and
Applied Mathematic
Some Progress in Conformal Geometry
This is a survey paper of our current research on the theory of partial
differential equations in conformal geometry. Our intention is to describe some
of our current works in a rather brief and expository fashion. We are not
giving a comprehensive survey on the subject and references cited here are not
intended to be complete. We introduce a bubble tree structure to study the
degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying
some global conformal bounds on compact manifolds of dimension 4. As
applications, we establish a gap theorem, a finiteness theorem for
diffeomorphism type for this class, and diameter bound of the
-metrics in a class of conformal 4-manifolds. For conformally compact
Einstein metrics we introduce an eigenfunction compactification. As a
consequence we obtain some topological constraints in terms of renormalized
volumes.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
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