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Hamiltonian stationary Lagrangian currents in Ricci-flat Kähler manifolds
Abstract
In this paper, we investigate Hamiltonian stationary Lagrangian currents with single-valued phases in open sets of Ricci-flat Kähler manifolds. We first establish the relation between phase and mean curvature for a class of integral Lagrangian currents in Ricci-flat Kähler manifolds. Then we derive several fundamental properties of integral Lagrangian currents with single-valued harmonic phases, including an almost monotonicity formula, an Allard-type regularity theorem, and the structure of limits of currents in this class. Finally, we apply the above results to study the regularity and rigidity of Hamiltonian stationary Lagrangian graphs in complex Euclidean space under certain conditions.
Related
- Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms
- Lagrangian Foliations on compact Kähler manifolds
- Hamiltonian stationary Lagrangian surfaces in complex space forms with constant-length gradient of the mean curvature
- Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
- Sharp scalar and Ricci curvature pinching theorems for Lagrangian submanifolds in the homogeneous nearly Kähler Six-sphere
BibTeX
@misc{ding2026hamiltonian,
title = {{Hamiltonian stationary Lagrangian currents in Ricci-flat Kähler manifolds}},
author = {Qi Ding},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14444},
eprint = {2609.14444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14444},
}
RIS
TY - UNPB
TI - Hamiltonian stationary Lagrangian currents in Ricci-flat Kähler manifolds
AU - Qi Ding
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14444
UR - https://arxiv.org/abs/2609.14444
AB - In this paper, we investigate Hamiltonian stationary Lagrangian currents with single-valued phases in open sets of Ricci-flat Kähler manifolds. We first establish the relation between phase and mean curvature for a class of integral Lagrangian currents in Ricci-flat Kähler manifolds. Then we derive several fundamental properties of integral Lagrangian currents with single-valued harmonic phases, including an almost monotonicity formula, an Allard-type regularity theorem, and the structure of limits of currents in this class. Finally, we apply the above results to study the regularity and rigidity of Hamiltonian stationary Lagrangian graphs in complex Euclidean space under certain conditions.
ER -