Year: 2026
Type: preprint
Abstract: In 2002, Montiel and Urbano conjectured that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore-type functional $ \mathcal{W}^-=\int_{T^2}(2+|H|^2)\,dA,$ either among all tori or among all Lagrangian tori. In this paper, we confirm this conjecture in the Lagrangian setting and disprove it in the general setting. We establish that every oriented closed Lagrangian surface of genus $g\geq 1$ in $\mathbb{C}P^2$ has $\mathcal{W}^-$-energy no less than that of the Clifford torus. Moreover, we construct non-Lagrangian deformations of the Clifford torus along which $\mathcal{W}^-$ strictly decreases.
Source: arXiv (Cornell University)
Authors: Wang, Peng, Xie, Zhenxiao, Zhao, Chen
Institutions: Fujian Normal University, Beihang University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1220981610, G3912457081
OpenAlex ID: W7214098395