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Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization. (English) Zbl 1378.35121

Let \(i:\;\Sigma \hookrightarrow (M,g)\) be an immersion of a closed (compact, without boundary) two-dimensional surface \(\Sigma\) into a Riemannian \(3\)-manifold \((M,g).\) The Willmore functional is defined by \[ W(i)=\int_\Sigma H^2 d\sigma, \] where \(d\sigma\) is the area form induced by the immersion and \(H\) is the mean curvature (the sum of the principal curvatures), that is, \(H=\bar{g}^{ij}A_{ij}\) is the trace of the second fundamental form \(A_{ij}\) with respect to the induced metric \(\bar{g}_{ij}.\) An immersion \(i\) is then called Willmore surface (or Willmore immersion) if it is a critical point of the Willmore functional with respect to normal perturbations or, equivalently, if it satisfies the associated Euler-Lagrange equation \[ \Delta_{\bar{g}}H + H|\overset{\circ}{A}|^2 + H \text{Ric}(n,n)=0, \] where \(\Delta_{\bar{g}}\) is the Laplace-Beltrami operator corresponding to the induced metric \(\bar{g},\) \((\overset{\circ}{A})_{ij}=A_{ij}-\frac{1}{2}H\bar{g}_{ij}\) is the trace-free second fundamental form, \(n\) is a normal unit vector to \(i,\) and \(\text{Ric}\) is the Ricci tensor of the ambient manifold \((M,g).\)
The very interesting paper under review is the first of a series of two papers devoted to the construction of embedded Willmore tori with small area constraint in Riemannian \(3\)-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction, and to this aim the authors exploit the variational structure of the problem in order to establish new geometric expansions of exponentiated small symmetric Clifford tori and to analyze the sharp asymptotic behaviour of degenerating tori under the action of the Möbius group.
The authors prove here two existence results by minimizing or maximizing a suitable reduced functional. More precisely, existence of embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) is obtained in compact \(3\)-manifolds with constant scalar curvature and in the double Schwarzschild space.

MSC:

35J60 Nonlinear elliptic equations
49Q10 Optimization of shapes other than minimal surfaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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