Lagrangian Translating Solitons and Special Lagrangians in $\mathbb{C}^m$ with Symmetries

TL;DR

This paper constructs new families of Lagrangian translating solitons and special Lagrangian submanifolds in complex Euclidean spaces, exhibiting symmetries under various group actions, expanding the known examples and classifications in the field.

Contribution

It introduces explicit constructions of symmetric Lagrangian solitons and submanifolds in C^m, generalizing previous methods and providing a full classification for certain group actions.

Findings

Explicit examples of symmetric Lagrangian translators and special Lagrangians.

Classification of admissible group actions for symmetry.

Construction of cohomogeneity-two and one examples.

Abstract

We construct novel families of exact immersed and embedded Lagrangian translating solitons and special Lagrangian submanifolds in $\mathbb{C}^m$ that are invariant under the action of various admissible compact subgroups $G \leq \text{SU}(m-1)$ with cohomogeneity-two. These examples are obtained via an Ansatz generalising a construction of Castro-Lerma in $\mathbb{C}^2$. We give explicit examples of admissible group actions, including a full classification for $G$ simple. We also describe novel Lagrangian translators symmetric with respect to non-compact subgroups of the affine special unitary group $\text{SU}(m)\ltimes \mathbb{C}^m$, including cohomogeneity-one examples.