Closed Lagrangian self-shrinkers in $\mathbb{R}^4$ symmetric with respect to a hyperplane

In this paper, we prove that the closed Lagrangian self-shrinkers in $\mathbb{R}^4$ which are symmetric with respect to a hyperplane are given by the products of Abresch–Langer curves. As a corollary, we obtain a new geometric characterization of the Clifford torus as the unique embedded closed Lagrangian self-shrinker symmetric with respect to a hyperplane in $\mathbb{R}^4$.

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This work was supported in part by NRF-2018R1A2B6004262.

Received 19 August 2020

Accepted 2 September 2021

Published 10 August 2024