31 research outputs found
Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization
We construct embedded Willmore tori with small area constraint in Riemannian
three-manifolds under some curvature condition used to prevent M\"obius
degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this
aim we establish new geometric expansions of exponentiated small symmetric
Clifford tori and analyze the sharp asymptotic behavior of degenerating tori
under the action of the M\"obius group. In this first work we prove two
existence results by minimizing or maximizing a suitable reduced functional, in
particular we obtain embedded area-constrained Willmore tori (or, equivalently,
toroidal critical points of the Hawking mass under area-constraint) in compact
3-manifolds with constant scalar curvature and in the double Schwarzschild
space. In a forthcoming paper new existence theorems will be achieved via Morse
theory.Comment: 41 pages. Final version to appear in the Proceedings of the London
Math. Societ
Henbunho o mochiita hi senkei renritsu Shuredinga hoteishikikei, hi senkei Sukara-ba hoteishiki no kenkyu
制度:新 ; 報告番号:甲3280号 ; 学位の種類:博士(理学) ; 授与年月日:2011/3/15 ; 早大学位記番号:新558
On weak solutions to a fractional Hardy-H\'enon equation: Part II: Existence
This paper and [29] treat the existence and nonexistence of stable weak
solutions to a fractional Hardy--H\'enon equation in , where , , and . In this paper, when is critical or supercritical in the
sense of the Joseph--Lundgren, we prove the existence of a family of positive
radial stable solutions, which satisfies the separation property. We also show
the multiple existence of the Joseph--Lundgren critical exponent for some and , and this property does not hold in the case
.Comment: 52 page
Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds II: Morse Theory
This is the second of a series of two papers where we construct embedded
Willmore tori with small area constraint in Riemannian three-manifolds. In both
papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty
being the M\"obius degeneration of the tori. In the first paper the
construction was performed via minimization, here by Morse Theory; to this aim
we establish new geometric expansions of the derivative of the Willmore
functional on exponentiated small Clifford tori degenerating, under the action
of the M\"obius group, to small geodesic spheres with a small handle. By using
these sharp asymptotics we give sufficient conditions, in terms of the ambient
curvature tensors and Morse inequalities, for having existence/multiplicity of
embedded tori stationary for the Willmore functional under the constraint of
prescribed (sufficiently small) area.Comment: Final version, to appear in the American Journal of Mathematic