Stabilities of affine Legendrian submanifolds and their moduli spaces

TL;DR

This paper introduces affine Legendrian submanifolds in Sasakian manifolds, defines a canonical volume, studies their stability, and explores the moduli space of special affine Legendrian submanifolds, revealing their geometric properties.

Contribution

It defines affine Legendrian submanifolds, introduces the $$-volume, derives stability results, and characterizes the moduli space of special affine Legendrian submanifolds.

Findings

Second variation formula establishes stability in $$-volume.

Convexity of the $$-volume functional on affine Legendrian submanifolds.

Moduli space of compact special affine Legendrian submanifolds is a smooth Fréchet manifold.

Abstract

We introduce the notion of affine Legendrian submanifolds in Sasakian manifolds and define a canonical volume called the $\phi$-volume as odd dimensional analogues of affine Lagrangian (totally real or purely real) geometry. Then we derive the second variation formula of the $\phi$-volume to obtain the stability result in some $\eta$-Einstein Sasakian manifolds. It also implies the convexity of the $\phi$-volume functional on the space of affine Legendrian submanifolds. Next, we introduce the notion of special affine Legendrian submanifolds in Sasaki-Einstein manifolds as a generalization of that of special Legendrian submanifolds. Then we show that the moduli space of compact connected special affine Legendrian submanifolds is a smooth Fr\'{e}chet manifold.