Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121120200301Weil-Petersson metrics on deformation spaces11712810418410.30504/jims.2020.104184ENH.-D.CaoDepartment of Mathematics, Lehigh University, Bethlehem, PA 18015, USA.X.SunDepartment of Mathematics, Lehigh University, Bethlehem, PA 18015, USA.S.-T.YauDepartment of Mathematics, Harvard University, Cambridge, MA 02138, USA.Y.ZhangYau mathematical Sciences Center, Tsinghua University, Beijing, 100804, China.Journal Article20200206In this paper we survey various aspects of the classical wpm and its generalizations, in particular on the moduli space of ke manifolds. Being a natural $L^2$ metric on the parameter space of a family of complex manifolds (or holomorphic vector bundles) which admit some canonical metrics, the wpm is well defined when the automorphism group of each fiber is discrete and the curvature of the wpm can be computed via certain integrals over each fiber. We shall discuss the Fano case when these fibers may have continuous automorphism groups. We also discuss the relation between the wpm on Teichm"uller spaces of K"ahler-Einstein manifolds of general type and energy of harmonic maps. <br /><br />In this paper we survey various aspects of the classical wpm and its generalizations, in particular on the moduli space of ke manifolds. Being a natural $L^2$ metric on the parameter space of a family of complex manifolds (or holomorphic vector bundles) which admit some canonical metrics, the wpm is well defined when the automorphism group of each fiber is discrete and the curvature of the wpm can be computed via certain integrals over each fiber. We shall discuss the Fano case when these fibers may have continuous automorphism groups. We also discuss the relation between the wpm on Teichm"uller spaces of K"ahler-Einstein manifolds of general type and energy of harmonic maps.http://jims.ims.ir/article_104184_b5c880cdd6543fe6c6b09d421a48be6a.pdf