TY - JOUR

T1 - An energy functional for Lagrangian tori in CP2

AU - Ma, Hui

AU - Mironov, Andrey E.

AU - Zuo, Dafeng

PY - 2018/6/1

Y1 - 2018/6/1

N2 - A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane (Formula presented.). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional (Formula presented.) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori.

AB - A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane (Formula presented.). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional (Formula presented.) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori.

KW - Energy functional

KW - Lagrangian surfaces

KW - Novikov–Veselov hierarchy

KW - Novikov-Veselov hierarchy

KW - SUBMANIFOLDS

KW - MINIMAL TORI

KW - EXAMPLES

KW - SURFACES

UR - http://www.scopus.com/inward/record.url?scp=85038405875&partnerID=8YFLogxK

U2 - 10.1007/s10455-017-9589-6

DO - 10.1007/s10455-017-9589-6

M3 - Article

AN - SCOPUS:85038405875

VL - 53

SP - 583

EP - 595

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 4

ER -