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

作者：Ikoma Norihisa^*; Malchiodi Andrea; Mondino Andrea

来源：Proceedings of the London Mathematical Society, 2017, 115(3): 502-544.

DOI：10.1112/plms.12047

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Mobius 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 Mobius 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.

• 出版日期2017-9