Ir al contenido

Documat

Balanced metrics, Zoll deformations and isosystolic inequalities in CPn

  • Luciano L. Junior [1]
    1. [1] Università di Trento, Trento, Italy
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 32, Nº. 2, 2026
  • Idioma: inglés
  • DOI: 10.1007/s00029-026-01152-y
  • Enlaces
  • Resumen

    • The k-systole of a Riemannian manifold is the infimum of the volume over all homologically non-trivial k-cycles. In this paper we discuss the behavior of the dimension two and co-dimension two systole of the complex projective space for distinguished classes of metrics, namely the homogeneous metrics and the balanced metrics. In particular, we argue that every homogeneous metric maximizes the systole in its volume-normalized conformal class, as well as that each Kähler metric locally minimizes the systole on the set of volume-normalized balanced metrics. The proof demands the implementation of integral geometric techniques, and a careful analysis of the second variation of the systole functional. As an application, we characterize the systolic behavior of almost-Hermitian 1-parameter Zoll-like deformations of the Fubini-Study metric.

  • Referencias bibliográficas
    • Pu, P.M.: Some inequalities in certain nonorientable Riemannian manifolds. Pac. J. Math. 2(1), 55–71 (1952)
    • Gromov, M.: Systoles and intersystolic inequalities. In: Broy, M., Denert, E. (eds.) Actes de la Table Ronde de Géométrie Différentielle (Luminy,...
    • Guth, L.: Metaphors in systolic geometry. In: Proceedings of the International Congress of Mathematicians. Volume II, pp. 745–768. Hindustan...
    • Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153, p. 676. Springer, New York (1969) Berger,...
    • Gromov, M.: Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82(2), 307–347 (1985). https://doi.org/10.1007/BF01388806
    • Katz, M.G., Suciu, A.I.: Volume of Riemannian manifolds, geometric inequalities, and homotopy theory. In: Tel Aviv Topology Conference: Rothenberg...
    • Sikorav, J.-C.: Dual elliptic planes. In: Actes des Journées Mathématiques à la Mémoire De Jean Leray. Sémin. Congr., vol. 9, pp. 185–207....
    • Zoll, O.: Über Flächen mit Scharen geschlossener geodätischer Linien. Math. Ann. 57, 108–133 (1903)
    • Besse, A.L.: Manifolds All of Whose Geodesics Are Closed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. Springer, Berlin, Heidelberg...
    • Guillemin, V.: The Radon transform on Zoll surfaces. Advances in Math. 22(1), 85–119 (1976). https://doi.org/10.1016/0001-8708(76)90139-0
    • Abbondandolo, A., Bramham, B., Hryniewicz, U.L., Salomão, P.A.S.: A systolic inequality for geodesic flows on the two-sphere. Math. Ann. 367,...
    • Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics. Springer, Berlin, Heidelberg (2005) ISBN: 978-3-540-26961-8,...
    • Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A 362(1711),...
    • Fu, J., Zhou, X.: Twistor geometry of Riemannian 4-manifolds by moving frames. Comm. Anal. Geom. 23(4), 819–839 (2015). https://doi.org/10.4310/CAG.2015.v23.n4.a4
    • Verbitsky, M.: Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds. Math. Res. Lett. 16(4), 735–752 (2009). https://doi.org/10.4310/MRL.2009.v16.n4.a14
    • Tomberg, A.: Twistor spaces of hypercomplex manifolds are balanced. Adv. Math. 280, 282–300 (2015). https://doi.org/10.1016/j.aim.2015.04.024
    • Fu, J.-X., Yau, S.-T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. Journal of Differential...
    • Yano, K., Mogi, I.: On real representations of Kaehlerian manifolds. Ann. Math. 61(1), 170–189 (1955). https://doi.org/10.2307/1969627 Article...
    • Hangan, T.: On the Riemannian metrics in which admit all hyperplanes as minimal hypersurfaces. J. Geom. Phys. 18(4), 326–334 (1996). https://doi.org/10.1016/0393-0440(95)00007-0
    • Hangan, T.: On Beltrami’s Theorem and generalized Heisenberg groups. Bollettino della Unione Matematica Italiana-B 2, 107–114 (1997)
    • Gray, A.: Minimal varieties and almost Hermitian submanifolds. Mich. Math. J. 12(3), 273–287 (1965). https://doi.org/10.1307/mmj/1028999364
    • Harvey, R., Lawson, H.B., Jr.: Calibrated geometries. Acta Math. 148, 47–157 (1982). https://doi.org/10.1007/BF02392726
    • Hirzebruch, F., Kodaira, K.: On the complex projective spaces. J. Math. Pures Appl. 9(36), 201–216 (1957)
    • Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. U.S.A. 74(5), 1798–1799 (1977). https://doi.org/10.1073/pnas.74.5.1798
    • Mazzucchelli, M., Suhr, S.: A characterization of Zoll Riemannian metrics on the 2-sphere. Bull. Lond. Math. Soc. 50(6), 997–1006 (2018)....
    • Ambrozio, L., Marques, F.C., Neves, A.: Rigidity theorems for the area widths of Riemannian manifolds (2024). https://arxiv.org/abs/2408.14375...
    • Ambrozio, L., Montezuma, R.: On the two-systole of real projective spaces. Rev. Mat. Iberoam. 36(7), 1979–1988 (2020). https://doi.org/10.4171/rmi/1188
    • Frölicher, A., Nijenhuis, A.: A theorem on stability of complex structures. Proc. Nat. Acad. Sci. U.S.A. 43, 239–241 (1957). https://doi.org/10.1073/pnas.43.2.239
    • Gray, A.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Annali di Matematica 123, 35–58 (1980). https://doi.org/10.1007/BF01796539
    • Nijenhuis, A., Woolf, W.B.: Some integration problems in almost-complex and complex manifolds. Ann. Math. 77(3), 424–489 (1963). https://doi.org/10.2307/1970126
    • Kruglikov, B.S.: Non-existence of higher-dimensional pseudoholomorphic submanifolds. Manuscripta Math. 111, 51–69 (2003). https://doi.org/10.1007/s00229-002-0352-2
    • Paiva, J.C., Fernandes, E.: Gelfand transforms and Crofton formulas. Selecta Math.(N.S.) 13(3), 369–390 (2007). https://doi.org/10.1007/s00029-007-0045-5
    • Morrey, C.B., Eells, J.: A variational method in the theory of harmonic integrals, i. Ann. Math. 63(1), 91–128 (1956). https://doi.org/10.2307/1969992
    • Huybrechts, D.: Complex Geometry: An Introduction. Universitext (Berlin. Print). Springer, Berlin, Heidelberg (2005). ISBN: 3-540-21290-6....
    • Ambrozio, L., Montezuma, R.: On the min-max width of unit volume three-spheres. J. Differential Geom. 126(3), 875–907 (2024). https://doi.org/10.4310/jdg/1717348867
    • Ogiue, K.: On compact complex submanifolds of the complex projective space. Tohoku Mathematical Journal 22(1), 95–97 (1970). https://doi.org/10.2748/tmj/1178242863
    • Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Wiley Classics Library. John Wiley & Sons Inc, New York (1996)
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno