A Uniqueness Theorem for Gluing Calibrated Submanifolds

TL;DR

This paper establishes a uniqueness theorem for higher-dimensional calibrated submanifolds, particularly special Lagrangian ones, constructed via the gluing technique in geometric analysis.

Contribution

It provides the first uniqueness result for higher-dimensional calibrated submanifolds obtained through gluing, extending the understanding of their geometric properties.

Findings

Proves a uniqueness theorem for glued calibrated submanifolds.

Extends the theory of calibrated geometry to higher dimensions.

Clarifies the conditions under which such submanifolds are uniquely determined.

Abstract

`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing technique. We have existence theorems in that sense, but there seems to have been no uniqueness theory for higher-dimensional ones such as special Lagrangian submanifolds, which we discuss in the present paper.