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April, 2020 On Hamiltonian stable Lagrangian tori in complex hyperbolic spaces
Toru KAJIGAYA
J. Math. Soc. Japan 72(2): 435-463 (April, 2020). DOI: 10.2969/jmsj/81158115

Abstract

In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n \leq 2$ and there exist infinitely many H-unstable $T^n$-orbits when $n \geq 3$. On the other hand, we prove a monotone $T^n$-orbit in $\mathbb{C}H^n$ is H-stable and rigid for any $n$. Moreover, we see almost all Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ are not Hamiltonian volume minimizing when $n \geq 3$ as well as the case of $\mathbb{C}^n$ and $\mathbb{C}P^n$.

Funding Statement

This work was supported by JSPS KAKENHI Grant Number JP18K13420.

Citation

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Toru KAJIGAYA. "On Hamiltonian stable Lagrangian tori in complex hyperbolic spaces." J. Math. Soc. Japan 72 (2) 435 - 463, April, 2020. https://doi.org/10.2969/jmsj/81158115

Information

Received: 23 August 2018; Published: April, 2020
First available in Project Euclid: 5 February 2020

zbMATH: 07196909
MathSciNet: MR4090343
Digital Object Identifier: 10.2969/jmsj/81158115

Subjects:
Primary: 53D12
Secondary: 53C42

Keywords: complex hyperbolic spaces , Hamiltonian stable Lagrangian submanifolds

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 2 • April, 2020
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