# $f$-minimal Lagrangian Submanifolds in K\"ahler Manifolds with Real Holomorphy Potentials

## TL;DR

This paper investigates the variational properties and stability of $f$-minimal Lagrangian submanifolds in K"ahler manifolds with real holomorphy potentials, including soliton solutions for Lagrangian mean curvature flow and their deformations.

## Contribution

It derives a second variation formula for $f$-minimal Lagrangians, generalizes known formulas, and introduces calibrated submanifolds in gradient steady K"ahler--Ricci solitons.

## Key findings

- Stability results for expanding and translating solitons in LMCF.
- Definition of $f$-volume calibrated submanifolds as generalizations of special Lagrangians.
- Noncompactness of these calibrated submanifolds.

## Abstract

The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in K\"ahler manifolds with real holomorphy potentials. Examples of submanifolds of this kind incuding soliton solutions for Lagrangian mean curvature flow (LMCF). We derive second variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh's formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady K\"ahler--Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF, and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.