Cheeger's inequalities for general symmetric forms and existence criteria for spectral gap

In this paper, some new forms of Cheeger's inequalities are established for general (maybe unbounded) symmetric forms (Theorems 1.1 and 1.2): the resu lting estimates improve and extend the ones obtained by Lawler and Sokal fo r bounded jump processes. Furthermore, some existence criteria for spectral gap of general symmetric forms or general reversible Markov processes are presented (Theorems 1.4 and 3.1), based on Cheeger's inequalities and a rel ationship between the spectral gap and the first Dirichlet and Neumann eige nvalues on local region.