Split Special Lagrangian Geometry

TL;DR

This paper explores split special Lagrangian geometry in the context of double numbers, establishing foundational results such as singularity removal, classification of cones, and connections to mass transport problems, highlighting both similarities and differences with classical complex geometry.

Contribution

It introduces the theory of split SLAG submanifolds in Hermitian D-manifolds, proves a removable singularities theorem, and links the geometry to optimal mass transport, expanding the understanding of semi-Riemannian calibrations.

Findings

No non-trivial split SLAG cones exist outside planes.

Split SLAG submanifolds are characterized by harmonic 1-forms.

Connection established between split SLAG geometry and Monge-Kantorovich problem.

Abstract

One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is replaced by tau with tau^2 = 1. A rather surprising amount of complex geometry carries over, almost untouched, and this has been the subject of many papers. We briefly review this material and, in particular, we discuss Hermitian D-manifolds with trivial canonical bundle, which provide the background space for the geometry of split SLAG submanifolds. A removable singularities result is proved for split SLAG subvarieties. It implies, in particular, that there exist no split SLAG cones, smooth outside the origin, other than planes. This is in sharp contrast to the complex case. Parallel to the complex case, space-like Lagrangian submanifolds are stationary if and only if they are theta-split SLAG for some phase angle theta, and infinitesimal deformations of split SLAG submanifolds are characterized by harmonic 1-forms on the submanifold. We also briefly review the recent work of Kim, McCann and Warren who have shown that split Special Lagrangian geometry is directly related to the Monge-Kantorovich mass transport problem.