Determine the global minimum of the Willmore-type functional among embedded tori

Determine the value of the infimum of the Willmore-type functional \(\mathcal{W}^-\) over all smooth embedded tori in \(\mathbb{C}P^2\), establish whether the infimum is attained by a smooth embedding, and, if it is attained, describe the geometry of the minimizer and determine whether the minimizing torus lies on a bifurcation branch generated by non-Killing Jacobi fields of the Clifford torus.

Background

The paper disproves the Montiel–Urbano conjecture for unrestricted smooth tori by constructing non-Lagrangian deformations of the Clifford torus $T_{CL}$ along which $\mathcal{W}^-$ strictly decreases. Consequently, the Clifford torus is not a minimizer of $\mathcal{W}^-$ in the class of all tori, although it is the minimizer in the Lagrangian class.

The authors therefore introduce $\beta^-$ as the infimum of $\mathcal{W}^-$ over smooth embeddings $F:T^2\to\mathbb{C}P^2$. The unresolved issues are the numerical value of this infimum, its attainment by a smooth embedded torus, the geometry of any minimizer, and its possible relation to bifurcation branches arising from the non-Killing Jacobi fields of the Clifford torus.