Boundary case of equality in optimal Loewner-type inequalities
HTML articles powered by AMS MathViewer
- by Victor Bangert, Christopher Croke, Sergei V. Ivanov and Mikhail G. Katz PDF
- Trans. Amer. Math. Soc. 359 (2007), 1-17 Request permission
Abstract:
We prove certain optimal systolic inequalities for a closed Riemannian manifold $(X,\mathcal {G})$, depending on a pair of parameters, $n$ and $b$. Here $n$ is the dimension of $X$, while $b$ is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from $X$ to its Jacobi torus $\mathbb {T}^b$, which are area-decreasing (on $b$-dimensional areas), with respect to suitable norms. These norms are the stable norm of $\mathcal {G}$, the conformally invariant norm, as well as other $L^p$-norms. Here we exploit $L^p$-minimizing differential 1-forms in cohomology classes. We characterize the case of equality in our optimal inequalities, in terms of the criticality of the lattice of deck transformations of $\mathbb {T}^b$, while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover.References
- H.W. Alt, Lineare Funktionalanalysis, 3, Auflage, Springer, 1999.
- Bernd Ammann, Dirac eigenvalue estimates on two-tori, J. Geom. Phys. 51 (2004), no. 3, 372–386. MR 2079417, DOI 10.1016/j.geomphys.2003.12.004
- I. Babenko, Géométrie systolique des variétés de groupe fondamental $\mathbb {Z}_2$, Sémin. Théor. Spectr. Géom. Grenoble, 22 (2004), 25-52.
- Paul Baird and John C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series, vol. 29, The Clarendon Press, Oxford University Press, Oxford, 2003. MR 2044031, DOI 10.1093/acprof:oso/9780198503620.001.0001
- V. Bangert, C. Croke, S. Ivanov, and M. Katz, Filling area conjecture and ovalless real hyperelliptic surfaces, Geom. Funct. Anal. 15 (2005), no. 3, 577–597. MR 2221144, DOI 10.1007/s00039-005-0517-8
- Victor Bangert and Mikhail Katz, Stable systolic inequalities and cohomology products, Comm. Pure Appl. Math. 56 (2003), no. 7, 979–997. Dedicated to the memory of Jürgen K. Moser. MR 1990484, DOI 10.1002/cpa.10082
- Victor Bangert and Mikhail Katz, An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm, Comm. Anal. Geom. 12 (2004), no. 3, 703–732. MR 2128608
- E. S. Barnes, On a theorem of Voronoi, Proc. Cambridge Philos. Soc. 53 (1957), 537–539. MR 86081, DOI 10.1017/s0305004100032527
- D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259–269. MR 1274115, DOI 10.1007/BF01896241
- D. Burago and S. Ivanov, On asymptotic volume of tori, Geom. Funct. Anal. 5 (1995), no. 5, 800–808. MR 1354290, DOI 10.1007/BF01897051
- Christopher B. Croke and Mikhail Katz, Universal volume bounds in Riemannian manifolds, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 109–137. MR 2039987, DOI 10.4310/SDG.2003.v8.n1.a4
- Albert Fathi and Antonio Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math. 155 (2004), no. 2, 363–388. MR 2031431, DOI 10.1007/s00222-003-0323-6
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Herbert Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767–771. MR 260981, DOI 10.1090/S0002-9904-1970-12542-3
- Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
- Jacqueline Ferrand, Sur la régularité des applications conformes, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 1, A77–A79 (French, with English summary). MR 470894
- Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
- Mikhael Gromov, Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 291–362 (English, with English and French summaries). MR 1427763
- Misha Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)]; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. MR 1699320
- C. Hamburger, Regularity of differential forms minimizing degenerate elliptic functionals, J. Reine Angew. Math. 431 (1992), 7–64. MR 1179331, DOI 10.1515/crll.1992.431.7
- S. V. Ivanov, On two-dimensional minimal fillings, Algebra i Analiz 13 (2001), no. 1, 26–38 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 1, 17–25. MR 1819361
- Sergei V. Ivanov and Mikhail G. Katz, Generalized degree and optimal Loewner-type inequalities, Israel J. Math. 141 (2004), 221–233. MR 2063034, DOI 10.1007/BF02772220
- Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR 0030135
- Mikhail Katz, Four-manifold systoles and surjectivity of period map, Comment. Math. Helv. 78 (2003), no. 4, 772–786. MR 2016695, DOI 10.1007/s00014-003-0774-9
- M. Katz, Systolic geometry and topology, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, to appear.
- Mikhail G. Katz and Christine Lescop, Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers, Geometry, spectral theory, groups, and dynamics, Contemp. Math., vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 181–200. MR 2180208, DOI 10.1090/conm/387/07242
- M. Katz and Y. Rudyak, Lusternik-Schnirelmann category and systolic category of low dimensional manifolds, Communications on Pure and Applied Mathematics, to appear. See arXiv:math.DG/0410456
- M. Katz and Y. Rudyak, Bounding volume by systoles of 3-manifolds. See arXiv:math.DG/0504008
- Mikhail G. Katz and Stéphane Sabourau, Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1189–1195. MR 2196056, DOI 10.1090/S0002-9939-05-08057-3
- Mikhail G. Katz and Stéphane Sabourau, Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1209–1220. MR 2158402, DOI 10.1017/S0143385704001014
- M. Katz and S. Sabourau, An optimal systolic inequality for CAT(0) metrics in genus two, Pacific J. Math., to appear. See arXiv:math.DG/0501017
- Lê Hông Vân, Curvature estimate for the volume growth of globally minimal submanifolds, Math. Ann. 296 (1993), no. 1, 103–118. MR 1213374, DOI 10.1007/BF01445097
- André Lichnerowicz, Applications harmoniques dans un tore, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A912–A916 (French). MR 253254
- Frank Morgan, Geometric measure theory, 2nd ed., Academic Press, Inc., San Diego, CA, 1995. A beginner’s guide. MR 1326605
- P.-A. Nagy, On length and product of harmonic forms in Kaehler geometry. See arXiv:math.DG/0406341
- Paul-Andi Nagy and Constantin Vernicos, The length of harmonic forms on a compact Riemannian manifold, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2501–2513. MR 2048527, DOI 10.1090/S0002-9947-04-03546-9
- P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71. MR 48886
- Peter Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150. MR 727034, DOI 10.1016/0022-0396(84)90105-0
- Brian White, The deformation theorem for flat chains, Acta Math. 183 (1999), no. 2, 255–271. MR 1738045, DOI 10.1007/BF02392829
Additional Information
- Victor Bangert
- Affiliation: Mathematisches Institut, Universität Freiburg, Eckerstrasse 1, 79104 Freiburg, Germany
- Email: bangert@mathematik.uni-freiburg.de
- Christopher Croke
- Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
- MR Author ID: 204906
- Email: ccroke@math.upenn.edu
- Sergei V. Ivanov
- Affiliation: Steklov Mathematics Institute, Fontanka 27, RU-191011 St. Petersburg, Russia
- MR Author ID: 337168
- Email: svivanov@pdmi.ras.ru
- Mikhail G. Katz
- Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan 52900, Israel
- MR Author ID: 197211
- Email: katzmik@math.biu.ac.il
- Received by editor(s): June 8, 2004
- Published electronically: August 24, 2006
- Additional Notes: The first author was partially supported by DFG-Forschergruppe “Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis”
The second author was supported by NSF grant DMS 02-02536 and the Max-Planck-Institut für Mathematik Bonn
The third author was supported by grants CRDF RM1-2381-ST-02, RFBR 02-01-00090, and NS-1914.2003.1
The fourth author was supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03) - © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 1-17
- MSC (2000): Primary 53C23; Secondary 57N65, 52C07
- DOI: https://doi.org/10.1090/S0002-9947-06-03836-0
- MathSciNet review: 2247879