Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: Minimization:

Norihisa Ikoma, Andrea Malchiodi, Andrea Mondino

Research output: Contribution to journalArticle

Abstract

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius 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 behaviour of degenerating tori under the action of the Möbius 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.

Original languageEnglish
Pages (from-to)502-544
Number of pages43
JournalProceedings of the London Mathematical Society
Volume115
Issue number3
DOIs
Publication statusPublished - 2017 Sep 1
Externally publishedYes

Fingerprint

Three-manifolds
Torus
Clifford Torus
Lyapunov-Schmidt Reduction
Constant Scalar Curvature
Morse Theory
Degeneration
Existence Theorem
Existence Results
Critical point
Asymptotic Behavior
Curvature

Keywords

  • 35J60
  • 49Q10
  • 53C21 (primary)
  • 53C42
  • 83C99 (secondary)

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I : Minimization: / Ikoma, Norihisa; Malchiodi, Andrea; Mondino, Andrea.

In: Proceedings of the London Mathematical Society, Vol. 115, No. 3, 01.09.2017, p. 502-544.

Research output: Contribution to journalArticle

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