5,772 research outputs found
On -superharmonic functions and some geometric applications
In this paper we study asymptotic behavior of -superharmonic functions at
isolated singularity using the Wolff potential and -capacity estimates in
nonlinear potential theory. Our results are inspired by and extend those of
Arsove-Huber and Taliaferro in 2 dimensions. To study -superharmonic
functions we use a new notion of -thinness by -capacity motivated by a
type of Wiener criterion in Arsove-Huber's paper. To extend Taliaferro's work,
we employ the Adams-Moser-Trudinger inequality for the Wolff potential, which
is inspired by the one used by Brezis-Merle. For geometric applications, we
study the asymptotic end behavior of complete conformally flat manifolds as
well as complete properly embedded hypersurfaces in hyperbolic space. In both
geometric applications the strong -capacity lower bound estimate of Gehring
in 1961 is brilliantly used. These geometric applications seem to elevate the
importance of -Laplace equations and make a closer tie to the classic
analysis developed in conformal geometry in general dimensions.Comment: 46 page
On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds
In this note we study constant mean curvature surfaces in asymptotically flat
3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive
mass, stable spheres of given constant mean curvature outside a fixed compact
subset are unique. Therefore we are able to conclude that there is a unique
foliation of stable spheres of constant mean curvature in an asymptotically
flat 3-manifold with positive mass.Comment: 22 page
On the topology of conformally compact Einstein 4-manifolds
In this paper we study the topology of conformally compact Einstein
4-manifolds. When the conformal infinity has positive Yamabe invariant and the
renormalized volume is also positive we show that the conformally compact
Einstein 4-manifold will have at most finite fundamental group. Under the
further assumption that the renormalized volume is relatively large, we
conclude that the conformally compact Einstein 4-manifold is diffeomorphic to
and its conformal infinity is diffeomorphic to .Comment: 16 page
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