Plurisubharmonic Functions in Calibrated Geometries

Plurisubharmonic functions in calibrated geometry and q-convexity

Let $(M,ω)$ be a Kahler manifold. An integrable function on M is called $ω^q$-plurisubharmonic if it is subharmonic on all q-dimensional complex subvarieties. We prove that a smooth $ω^q$-plurisubharmonic function is q-convex. A continuous $ω^q$-plurisubharmonic function admits a local approximation by smooth, $ω^q$-plurisubharmonic functions. For any closed subvariety $Z\subset M$, $\dim Z < q$, there exists a strictly $ω^q$-plurisubharmonic function in a neighbourhood of $Z$ (this result is known for q-convex functions). This theorem is used to give a new proof of Sibony's lemma on integrabi…

Duality of Positive Currents and Plurisubharmonic Functions in Calibrated Geometry

Recently the authors showed that there is a robust potential theory attached to any calibrated manifold (X,ϕ). In particular, on X there exist ϕ-plurisubharmonic functions, ϕ-convex domains, ϕ-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the ϕ-submanifolds, or more generally, the ϕ-currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the ϕ-convex hull of a compa…

An Introduction to Potential Theory in Calibrated Geometry

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. In particular, the notion of pseudo-convexity for a calibrated manifold (X,ϕ) is introduced and studied. Analogues of totally real submanifolds are also int…

校准几何中的刘维尔定理

Liouville's theorem in calibrated geometries

本文将经典刘维尔定理推广到校准几何中，证明在多数维数下校准具有刘维尔性质，并刻画例外情形，还证明具有该性质的校准构成稠密 $G_δ$ 集，并应用于拟正则曲线的分解与稳定性。

整共形曲线要么是仿射的要么具有超欧几里得能量增长

Entire conformal curves are affine or have super-Euclidean energy growth

该研究证明高维整共形曲线仅分为仿射曲线和超欧几里得能量增长两类，推导了相关刚性结论与应用，并给出了子流形共形等价的相关结果。