Continuity Equation of Transverse Kähler Metrics on Sasakian Manifolds

We introduce the continuity equation of transverse K & auml;hler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique eta-Einstein metric in the basic Bott-Chern cohomological class of the initial transverse K & auml;hler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the K & auml;hler metrics studied by La Nave and Tian, and also counterparts of the Sasaki-Ricci flow studied by Smoczyk, Wang, and Zhang.