H. Blaine Lawson

An introduction to potential theory in calibrated geometry

In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper introduces and investigates questions of pseudo-convexity in the context of a general calibrated manifold

Split Special Lagrangian Geometry

(X,\phi)

Duality of positive currents and plurisubharmonic functions in calibrated geometry

. Analogues of totally real submanifolds are introduced and used to construct enormous families of strictly

Finite volume flows and Morse theory

\phi

Algebraic cycles and the classical groups. I: real cycles

-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout. In a sequel, the duality between

Boundaries of varieties in projective manifolds

\phi

Tangents to Subsolutions Existence and Uniqueness, II

-pluri\-sub\-harmonic functions and

Algebraic cycles and the classical groups II: quaternionic cycles.

\phi

Codimension-one foliations of spheres

-positive currents is investigated. This study involves boundaries, generalized Jensen measures, and other geometric objects on a calibrated manifold.

Lagrangian potential theory and a Lagrangian equation of Monge–Ampère type

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 – = -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 Ѳ-split SLAG for some constant phase angle Ѳ, 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.