In this paper, we introduce the first-order differential operators d0 and d1 acting on the quaternionic version of differential forms on the flat quaternionic space H n . The behavior of d0, d1 and △ = d0d1 is very similar to ∂, ∂ and ∂∂ in several complex variables. The quaternionic Monge-Ampère operator can be defined as (△u) n and has a simple explicit expression. We define the notion of closed positive currents in the quaternionic case, and extend several results in complex pluripotential theory to the quaternionic case: define the Lelong number for closed positive currents, obtain the quaternionic version of Lelong-Jensen type formula, and generalize Bedford-Taylor theory, i.e., extend the definition of the quaternionic Monge-Ampère operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.
In this paper, we prove that for every Finsler n-sphere (S n , F ) for n ≥ 3 with reversibility λ and flag curvature K satisfying λ λ+1 2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiµ) with an irrational µ. Furthermore, there always exist three prime closed geodesics on any (S 3 , F ) satisfying the above pinching condition.
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