Reductions of minimal Lagrangian submanifolds with symmetries

TL;DR

This paper establishes a relationship between the minimality of $K$-invariant Lagrangian submanifolds in Fano manifolds and their reductions in K"ahler quotients, providing new examples of minimal Lagrangians through symmetry reductions.

Contribution

It proves the equivalence of minimality between original and reduced Lagrangian submanifolds under symmetry actions and offers explicit examples using circle actions on K"ahler-Einstein manifolds.

Findings

Minimality of $K$-invariant Lagrangians corresponds to minimality of reduced submanifolds.

Provides explicit examples of minimal Lagrangians via symmetry reduction.

Connects minimality in Fano manifolds with K"ahler quotient geometry.

Abstract

Let $M$ be a Fano manifold equipped with a K\"ahler form $\omega\in 2\pi c_1(M)$ and $K$ a connected compact Lie group acting on $M$ as holomorphic isometries. In this paper, we show the minimality of a $K$-invariant Lagrangian submanifold $L$ in $M$ w.r.t. a globally conformal K\"ahler metric is equivalent to the minimality of the reduced Lagrangian submanifold $L_0=L/K$ in a K\"ahler quotient $M_0$ w.r.t. the Hsiang-Lawson metric. Furthermore, we give some examples of K\"ahler reductions by using a circle action obtained from a cohomogenenity one action on a K\"ahler-Einstein manifold of positive Ricci curvature. Applying these results, we obtain several examples of minimal Lagrangian submanifolds via reductions.