Embedded area-constrained Willmore Tori of small area in Riemannian three-manifolds, II

Morse theory

Norihisa Ikoma, Andrea Malchiodi, Andrea Mondino

Research output: Contribution to journalArticle

Abstract

This is the second part 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 Mübius 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 small Clifford tori (in geodesic normal coordinates) which degenerate to small geodesic spheres with a small handle under the action of the Mübius group. 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 which are stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.

Original languageEnglish
Pages (from-to)5-1378
Number of pages1374
JournalAmerican Journal of Mathematics
Volume139
Issue number5
DOIs
Publication statusPublished - 2017 Jan 1
Externally publishedYes

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Three-manifolds
Morse Theory
Torus
Clifford Torus
Morse Inequalities
Geodesic Sphere
Lyapunov-Schmidt Reduction
Curvature Tensor
Degeneration
Geodesic
Multiplicity
Derivative
Series
Sufficient Conditions

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Embedded area-constrained Willmore Tori of small area in Riemannian three-manifolds, II : Morse theory. / Ikoma, Norihisa; Malchiodi, Andrea; Mondino, Andrea.

In: American Journal of Mathematics, Vol. 139, No. 5, 01.01.2017, p. 5-1378.

Research output: Contribution to journalArticle

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