Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121120200301Shing-Tung Yau's work on the notion of mass in general relativity1310526510.30504/jims.2020.105265ENM.-T.WangDepartment of Mathematics,
Columbia University, 2990 Broadway, New York, NY 10027.Journal Article20200321The notion of mass or energy has been one of the most challenging problems in general relativity since Einstein's time. As is well known from the equivalence principle, there is no well-defined concept of energy density for gravitation. On the other hand, when there is asymptotic symmetry, concepts of total energy and momentum can be defined. This is the ADM energy-momentum and the Bondi energy-momentum when the system is viewed from spatial infinity and null infinity, respectively. These concepts are fundamental in general relativity but there are limitations to such definitions if the physical system is not isolated and cannot quite be viewed from infinity where asymptotic symmetry exists.<br /><br />The positive energy conjecture states that the total energy of a nontrivial isolated physical system must be positive. This conjecture lies in the foundation of general relativity upon which stability of the system rests.This long standing conjecture had attracted many physicists and mathematician,<br />but only very special cases were verified up until the seventies.Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121120200301Moduli of $J$-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an $S^1$-equivariant pair59510418510.30504/jims.2020.104185ENC.C. MelissaLiuDepartment of Mathematics, Columbia University,
2990 Broadway, New York, NY 10027.Journal Article20200206Let $(X,omega)$ be a symplectic manifold, $J$ be an $omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized $J$-holomorphic curves in $X$ with boundary in $L$ (with prescribed topological data) is compact and Hausdorff in Gromov's $C^infty$-topology. We construct a Kuranishi structure with corners in the sense of Fukaya and Ono. This Kuranishi structure is orientable if $L$ is spin. In the special case where the expected dimension of the moduli space is zero, and there is an $S^1$-action on the pair $(X,L)$ which preserves $J$ and has no fixed points on $L$, we define the Euler number for this $S^1$-equivariant pair and the prescribed topological data. We conjecture that this rational number is the one computed by localization techniques using the given $S^1$-action.Iranian Mathematical SocietyJournal of the Iranian Mathematical Society2717-16121120200301Canonical sections of Hodge bundles on moduli spaces9711510418610.30504/jims.2020.104186ENK.LiuDepartment of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555, USAY.ShenDepartment of Mathematics, Nanjing University, Nanjing 210093, P.R.ChinaJournal Article20200206We review recent works in [K. Liu, S. Rao, and X. Yang, Quasi-isometry and deformations of Calabi Yau manifolds, Inventiones mathematicae, 199(2) (2015), 423–453] and [K. Liu and Y. Shen, Moduli spaces as ball quotients I, local theory, preprint] on geometry of sections of Hodge bundles and their applications to moduli spaces.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.