Calculus of Variations and Geometric Measure Theory
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N. Ikoma - A. Malchiodi - A. Mondino

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

created by mondino on 15 Oct 2014
modified on 06 May 2017


Proceedings of the London Math. Soc.

Inserted: 15 oct 2014
Last Updated: 6 may 2017

Year: 2014


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.


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