| Hodge theory is one of the most significant advances in mathematics in the 20th century.The famous Hodge theorem shows that the space of all harmonic r-forms on compact oriented Riemannian manifold M is a finite dimensional vector space,which is isomorphic to the rth de Rham cohomology group.The introduction of Hodge theory further reveals the deeper connection between analysis and topology,which has a great influence on the whole study of analysis of modern manifolds.Green operator is an integral operator defined on harmonic complement space,which plays a significant role in differential forms,especially in Hodge theory.In this paper,we mainly focus on some applications of Green operators to Hodge theory on(real)differential and complex manifolds.Firstly,over the space of differential forms of(real)differential and complex manifolds,we introduce three Green operator G.and harmonic projection operators H.associated to three Laplace operators △d、△(?)、△(?),and point out that the three versions of Hodge decomposition theorem can formulate such a unified equation Id=H.+△,G,.Furthermore,we show that every corresponding Green operator is a bounded self-adjoint elliptic operator from the view piont of d-、(?)-、(?)-Hodge decomposition theorem,respectively,and that there are isomorphic relations between harmonic space and de Rham cohomology group(resp.Dolbeault cohomology group).In addition,we also discuss the eigenvalue problem and commutativity of Green operators G..Secondly,note that the proof of Hodge decomposition theorem can be changed into solving Laplace equation on compact manifold,which is essentially to find a weak solution and then prove the smooth regularity of the solution.To this end,We use the theory of Sobolev space combining with the Green operator to discuss the weak problem and regularity.For the completeness,we also reproved the Garding inequality.Finally,motivated by solving equations with Hodge theory from Liu and Zhu’s work,we introduce two extended Green operators(?)*(?)and(?),and prove that they are of quasi-isometry in order to full display the iterative applications of Hodge decomposition theorem of(?)、(?)-version and properties of Green operator in the proving process.Furthermore,we discuss the role of the extended Green operator(?)in solving the(?)-differential equation(?)(iφΩ)0 which plays an important role in inspiring to solve the simplified equation of Siu’s conjecture. |
