28 research outputs found
Deformation of LeBrun's ALE metrics with negative mass
In this article we investigate deformations of a scalar-flat K\"ahler metric
on the total space of complex line bundles over CP^1 constructed by C. LeBrun.
In particular, we find that the metric is included in a one-dimensional family
of such metrics on the four-manifold, where the complex structure in the
deformation is not the standard one.Comment: 20 pages, no figure. V2: added two references, filled a gap in the
proof of Theorem 1.2. V3: corrected a wrong statement about Kuranishi family
of a Hirzebruch surface stated in the last paragraph in the proof of Theorem
1.2, and fixed a relevant error in the proof. Also added a reference [24]
about Kuranishi family of Hirzebruch surface
Asymptotic behavior of solutions to the -Yamabe equation near isolated singularities
-Yamabe equations are conformally invariant equations generalizing
the classical Yamabe equation. In an earlier work YanYan Li proved that an
admissible solution with an isolated singularity at to the
-Yamabe equation is asymptotically radially symmetric. In this work
we prove that an admissible solution with an isolated singularity at to the -Yamabe equation is asymptotic to a radial
solution to the same equation on . These results
generalize earlier pioneering work in this direction on the classical Yamabe
equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli
et al, we formulate and prove a general asymptotic approximation result for
solutions to certain ODEs which include the case for scalar curvature and
curvature cases. An alternative proof is also provided using
analysis of the linearized operators at the radial solutions, along the lines
of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page