Deformations of special Legendrian submanifolds in Sasaki-Einstein manifolds

TL;DR

This paper develops a deformation theory for special Legendrian submanifolds in Sasaki-Einstein manifolds, revealing conditions for unobstructed deformations and the existence of non-special Legendrian deformations via harmonic 1-forms.

Contribution

It introduces a new deformation framework for these submanifolds and shows their deformation space is an intersection of larger smooth spaces, including non-special Legendrian deformations.

Findings

Deformation space is the intersection of two larger smooth spaces.

Special Legendrian deformations can be obstructed.

Existence of smooth non-special Legendrian deformations via harmonic 1-forms.

Abstract

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, which are not special Legendrian deformations, given by harmonic $1$-forms.