2019/09/28 by Wei Wang, Wang, Wei
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research
paper · pdf · doi:10.48550/arxiv.1909.13109
Many fundamental results of pluripotential theory on the quaternionic space ℍn are extended to the Heisenberg group. We introduce notions of a plurisubharmonic function, the quaternionic Monge-Ampère operator, differential operators d0 and d1 and a closed positive current on the Heisenberg group. The quaternionic Monge-Ampère operator is the coefficient of (d0d1u)n. We establish the Chern-Levine-Nirenberg type estimate, the existence of quaternionic Monge-Ampère measure for a continuous quaternionic plurisubharmonic function and the minimum principle for the quaternionic Monge-Ampère operator. Unlike the tangential Cauchy-Riemann operator ∂b on the Heisenberg group which behaves badly as ∂b∂b≠ -∂b∂b , the quaternionic counterpart d0 and d1 satisfy d0d1=-d1d0 . This is the main reason that we have a better theory for the quaternionic Monge-Ampère operator than (∂b∂b)n.