ORIGINAL_ARTICLE Shing-Tung Yau's work on the notion of mass in general relativity The 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.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,but only very special cases were verified up until the seventies. http://jims.ims.ir/article_105265_9b45902aae90b22ffdb5c61d4bcb0aa2.pdf 2020-03-01 1 3 10.30504/jims.2020.105265 Mass General Relativity Positive energy conjecture M.-T. Wang mtwang@math.columbia.edu 1 Department of Mathematics‎, ‎Columbia University‎, ‎2990 Broadway‎, ‎New York‎, ‎NY 10027. LEAD_AUTHOR R. Schoen and S.-T. Yau, Positivity of the total mass of a general space-time, Phys. Rev. Lett. 43 (1979), no. 20, 1457--1459. 1 R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45--76. 2 R. Schoen and S.-T. Yau, Proof of the positive mass theorem II, Comm. Math. Phys. 79 (1981), no. 2, 231--260. 3 R. Schoen and S.-T. Yau, Proof that the Bondi mass is positive, Phys. Rev. Lett. 48 (1982), no. 6, 369--371. 4 R. Schoen and S.-T. Yau, The existence of a black hole due to condensation of matter, Comm. Math. Phys. 90 (1983), no. 4, 575--579 5 G. Huisken and S.-T. Yau, Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math. 124 (1996), no. 1-3, 281--311. 6 S.-T. Yau, Geometry of three manifolds and existence of black hole due to boundary effect, Adv. Theor. Math. Phys. 5 (2001), no. 4, 755--767. 7 M.-T. Wang and S.-T. Yau, Quasilocal mass in general relativity, Phys. Rev. Lett. 102 (2009), no. 2, no. 021101, 4 pages. 8 M.-T. Wang and S.-T. Yau, Isometric embeddings into the Minkowski space and new quasi-local mass, Comm. Math. Phys. 288 (2009), no. 3, 919--942. 9
ORIGINAL_ARTICLE Moduli of $J$-holomorphic curves with Lagrangian boundary conditions ‎and open Gromov-Witten invariants for an $S^1$-equivariant pair Let $(X,\omega)$ 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. http://jims.ims.ir/article_104185_c86f22dd64c9c7ae78d34f83e9d829f4.pdf 2020-03-01 5 95 10.30504/jims.2020.104185 Moduli of $J$-Holomorphic Curves Lagrangian boundary conditions ‎Open Gromov-Witten Invariants C.C. Melissa Liu ccliu@math.columbia.edu 1 ‎Department of Mathematics‎, ‎Columbia University‎, ‎2990 Broadway‎, ‎New York‎, ‎NY 10027. LEAD_AUTHOR W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, 820, Springer-Verlag, Berlin-New York, 1980. 1 N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957) 235--249. 2 N. L. Alling and N. Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics, 219, Springer-Verlag, Berlin-New York, 1971. 3 M. Aganagic, A. Klemm, and C. Vafa, Disk instantons, mirror symmetry, and the duality web, Z. Naturforsch. A 57 (2002), no. 1-2, 1--28 4 M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041. 5 E. Bujalance, J. J. Etayo, J. M. Gamboa, and G. Gromadzki, Automorphism groups of compact bordered Klein surfaces, Lecture Notes in Mathematics, 1439, Springer-Verlag, Berlin-New York, 1990. 6 K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45--88. 7 P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble supercon formal field theory, Nuclear Phys. B 359 (1991), no.1, 21--74. 8 K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), no. 5, 933--1048. 9 K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory--anomaly and obstruction, preprint 2000. 10 A. Givental, Equivariant Gromov-Witten invariants, Internat. Math. Res. Notices 1996, no. 13, 613--663. 11 R. Gopakumar and C. Vafa, M-theory and topological strings--II, hep-th/9812127. 12 T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487-518 13 T. Graber, E. Zaslow, Open-string Gromov-Witten invariants: Calculations and a Mirror theorem", Orbifolds in mathematics and physics (Madison, WI, 2001), 107--121, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002. 14 M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307--347. 15 R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977. 16 M. W. Hirsch Differential topology, Graduate Texts in Mathematics, 33, Springer-Verlag, New York-Heidelberg, 1976. 17 J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, 187, Springer-Verlag, New York, 1998. 18 S. Ivashkovich and V. Shevchishin, Gromov compactness theorem for J-complex curves with boundary, Internat.Math. Res. Notices 2000, no. 22, 1167--1206. 19 S. Ivashkovich and V. Shevchishin, Holomorphic structure on the space of Riemann surfaces with marked boundary, Tr. Mat. Inst. Steklova 235 (2001), Anal. i Geom. Vopr. Kompleks. Analiza, 98--109; reprinted in in Proc. Steklov Inst. Math. 2001 (235), no. 4, 91--102. 20 S. Ivashkovich, V. Shevchishin, Reection principle and J-complex curves with boundary on totally real immersions, Commun. Contemp. Math. 4 (2002), no. 1, 65--106. 21 F. Klein, Uber Realitatsverhaltnisse bei der einem beliebigen Geschlechte zugehorigen Normalkurve der φ, Math. Ann. 42 (1893), no. 1, 1-29. 22 K. Kodaira, Complex manifolds and deformation of complex structures, Grundlehrender Mathematischen Wissenschaften 283. Springer-Verlag, New York, 1986. 23 S. Katz and C. C. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), no. 1, 1--49. 24 B. Lian, K. Liu and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997), no. 4, 729--763. 25 J. Li and Y. S. Song, Open string instantons and relative stable morphisms, Adv. Theor. Math. Phys. 5 (2001), no. 1, 67--91. 26 J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119--174. 27 J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47--83, First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 1998. 28 G. Liu and G. Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998), no. 1, 1--74. 29 J. M. F. Labastida, M. Marino and C. Vafa Knots, links and branes at large N, Jour. High En. Phys. (2000), no. 11, Paper 7, 42 pp. 30 D. McDuff and D. Salamon, J-holomorphic curves and quantum cohomology, University Lecture Series, 6, American Mathematical Society, Providence, RI, 1994. 31 M. Mari~no and C. Vafa Framed knots at large N, Orbifolds in mathematics and physics (Madison, WI, 2001), 185--204, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002. 32 H. Ooguri and C. 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ORIGINAL_ARTICLE Canonical sections of Hodge bundles on moduli spaces We 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. http://jims.ims.ir/article_104186_5aeacde0efd83615df987f55d36727c2.pdf 2020-03-01 97 115 10.30504/jims.2020.104186 CANONICAL SECTIONS HODGE BUNDLES MODULI SPACES K. Liu liu@math.ucla.edu 1 Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095-1555, USA LEAD_AUTHOR Y. Shen syliuguang2007@163.com 2 Department of Mathematics, Nanjing University, Nanjing 210093, P.R.China AUTHOR D. Allcock, J. Carlson and D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Alg. Geom. 11 (2002), no. 4, 659-724. 1 D. Allcock, J. Carlson and D. Toledo, The Moduli Space of Cubic Threefolds as a Ball Quotient, Mem. Amer. Math. Soc. 209 (2011), no. 985, xii+70. 2 A. Beauville, Moduli of cubic surfaces and Hodge theory (after Allcock, Carlson, Toledo), G_eom_etriesa courbure negative ou nulle, groupes discrets et rigidit_es, S_eminaires et Congres 18 Soc. Math. France, Paris, 2009. 3 C. H. Clemens, Degenerations of Kahler manifolds, Duke Math J. 44 (1977), no. 2, 215--290. 4 C. H. Clemens, Geometry of formal Kuranishi theory, Adv. Math. 198 (2005), no. 1, 311--365. 5 P. Deligne, Th_eorie de Hodge II, Publ. Math. IHES 40 (1971) 5--57. 6 P. Deligne and G. W. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES 63 (1972) 5--89. 7 I. V. Dolgachev and S. Kondo, Moduli of K3 surfaces and complex ball quotients, Arithmetic and Geometry Around 8 Hypergeometric Functions, Progress in Mathematics 260, 2007, pp. 43-100. 9 P. Griffiths, Periods of integrals on algebraic manifolds I, Construction and properties of the modular varieties, Amer. J. Math. 90 (1968) 568--626. 10 P. Griffiths, Periods of integrals on algebraic manifolds II, Amer. J. Math. 90 (1968) 805--865. 11 P. Griffiths, On the Periods of Certain Rational Integrals: I, II, Ann. of Math. (2) 90 (1969), no. 2, 460--495 and 496--541. 12 P. Griffiths, Periods of integrals on algebraic manifolds, III, Some global differential-geometric properties of the period mapping, Publ. Math. IHES 38 (1970) 125--180. 13 P. Griffiths, Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems, Bull. Amer. Math. Soc. 76 (1970), no.2, 228--296. 14 P. Griffiths, Topics in transcendental algebraic geometry, Proceedings of a seminar held at the Institute for Advanced 15 Study, Princeton, N.J., during the academic year 1981/1982. Edited by Phillip Griffiths, Annals of Mathematics 16 Studies, 106. Princeton University Press, Princeton, NJ, 1984. 17 P. Griffiths and W. Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969) 253--302. 18 P. Griffiths and W. Schmid, Recent developments in Hodge theory: a discussion of techniques and results, Discrete 19 subgroups of Lie groups and applicatons to moduli (Internat. Colloq., Bombay, 1973), pp. 31--127. Oxford Univ. Press, Bombay, 1975. 20 P. Griffiths and J. Wolf, Complete maps and differentiable coverings, Michigan Math. J. 10 (1963) 253--255. 21 K. Kodaira and D. C. Spencer, On Deformations of Complex Analytic Structures, III, Ann. of Math. (2) 71 (1960) 43--76. 22 K. Liu, S. Rao and X. Yang, Quasi-isometry and deformations of CalabiYau manifolds, Invent. Math. 199 (2015), 23 no. 2, 423--453. 24 K. Liu and Y. Shen, Global Torelli theorem for projective manifolds of Calabi--Yau type, arXiv:1205.4207, (2012). 25 K. Liu and Y. Shen, Simultaneous normalization of period map and affine structures on moduli spaces, arXiv:1910.06767, (2019). 26 K. Liu and Y. Shen, From local Torelli to global Torelli, arXiv: 1512.08384, (2015). 27 K. Liu and Y. Shen, Moduli spaces as ball quotients I, local theory. preprint. 28 W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973) 211--319. 29 A. Sommese, On the rationality of the period mapping, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 4, 683--717. 30 B. Szendroi, Some finiteness results for Calabi-Yau threefolds, J. London Math. Soc. (2) 60 (1999), no. 3, 689--699. 31 G. Tian, Smoothness of the universal deformation space of compact Calabi--Yau manifolds and its PeterssonWeil metric, Mathematical aspects of string theory, In: Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987, pp. 629-646. 32 A. Todorov, The Weil-Petersson geometry of the moduli space of SU(n_3) (Calabi-Yau) manifolds. I, Commun. Math. Phys. 126 (1989), no. 2, 325--346. 33
ORIGINAL_ARTICLE Weil-Petersson metrics on deformation spaces 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. 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 2020-03-01 117 128 10.30504/jims.2020.104184 ‎Weil-Petersson metrics Deformation Spaces moduli space H.-D. Cao huc2@lehigh.edu 1 Department of Mathematics‎, ‎Lehigh University‎, ‎Bethlehem‎, ‎PA 18015‎, ‎USA. LEAD_AUTHOR X. Sun xis205@lehigh.edu 2 Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. AUTHOR S.-T. Yau yau@math.harvard.edu 3 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA. AUTHOR Y. Zhang zhangyingying@math.tsinghua.edu.cn 4 Yau mathematical Sciences Center, Tsinghua University, Beijing, 100804, China. AUTHOR L. Ahlfors, Some remarks on Teichmüller's space of Riemann surfaces, Ann. of Math. (2) 74 (1961) 171--191. 1 L. Ahlfors, Curvature properties of Teichmüller space. J. Analyse Math. 9 (1961/1962), 161--176. 2 S. Bando and T. Mabuchi, Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic geometry, Sendai, 1985, 11--40, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. 3 T. Brönnle, Extremal Kähler metrics on projectivized vector bundles, Duke Math. J. 164 (2015), no. 2, 195--233. 4 E. Calabi, Extremal Kähler metrics. II. Differential geometry and complex analysis, 95--114, Springer, Berlin, 1985. 5 P. 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