Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 1 1 2020 01 01 Weil-Petersson metrics on deformation spaces 117 128 104184 10.30504/jims.2020.104184 EN H.-D. Cao Department of Mathematics‎, ‎Lehigh University‎, ‎Bethlehem‎, ‎PA 18015‎, ‎USA. X. Sun Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. S.-T. Yau Department of Mathematics, Harvard University, Cambridge, MA 02138, USA. Y. Zhang Yau mathematical Sciences Center, Tsinghua University, Beijing, 100804, China. Journal Article 2020 02 06 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. <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. https://jims.ims.ir/article_104184_d7c4df52cd8c26ee902bce2a32b089a2.pdf