Boundary case of equality in optimal Loewner-type inequalities
Authors:
Victor Bangert, Christopher Croke, Sergei V. Ivanov and Mikhail G. Katz
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
Published electronically:
August 24, 2006
MathSciNet review:
2247879
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We prove certain optimal systolic inequalities for a closed Riemannian manifold , depending on a pair of parameters,
and
. Here
is the dimension of
, while
is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from
to its Jacobi torus
, which are area-decreasing (on
-dimensional areas), with respect to suitable norms. These norms are the stable norm of
, the conformally invariant norm, as well as other
-norms. Here we exploit
-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
, 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.
- [Al99] H.W. Alt, Lineare Funktionalanalysis, 3, Auflage, Springer, 1999.
- [Am04] B. Ammann, Dirac eigenvalue estimates on two-tori, J. Geom. Phys. 51 (2004), no. 3, 372-386. MR 2079417 (2005f:53068)
- [Bab04]
I. Babenko, Géométrie systolique des variétés de groupe fondamental
, Sémin. Théor. Spectr. Géom. Grenoble, 22 (2004), 25-52.
- [BaW03] I. Baird and John C. Wood, Harmonic morphisms between Riemannian manifolds, London Mathematical Society Monographs. New Series 29, The Clarendon Press, Oxford University Press, Oxford, 2003. MR 2044031 (2005b:53101)
- [BCIK04] V. Bangert, C. Croke, S. Ivanov, M. Katz, Filling area conjecture and ovalless real hyperelliptic surfaces, Geometric and Functional Analysis (GAFA) 15 (2005), no. 3, 577-597. See arXiv:math.DG/0405583 MR 2221144
- [BK03] V. Bangert and M. Katz, Stable systolic inequalities and cohomology products, Comm. Pure Appl. Math. 56 (2003), 979-997. Available at arXiv:math.DG/0204181 MR 1990484 (2004g:53047)
- [BK04] V. Bangert and M. Katz, An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm, Comm. Anal. Geom. 12 (2004), no. 3, 703-732. See arXiv:math.DG/0304494 MR 2128608
- [Bar57] E. S. Barnes, On a theorem of Voronoi, Proc. Cambridge Philos. Soc. 53 (1957), 537-539. MR 0086081 (19:120g)
- [BI94] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259-269. MR 1274115 (95h:53049)
- [BI95] D. Burago and S. Ivanov, On asymptotic volume of tori, Geom. Funct. Anal. 5 (1995), no. 5, 800-808. MR 1354290 (96h:53041)
- [CK03] C. Croke and M. Katz, Universal volume bounds in Riemannian manifolds, Surveys in Differential Geometry VIII, Lectures on Geometry and Topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, May 3-5, 2002 (S.T. Yau, ed.), Somerville, MA: International Press, 2003, pp. 109-137. See arXiv:math.DG/0302248 MR 2039987 (2005d:53061)
- [FaS04]
A. Fathi and A. Siconolfi, Existence of
critical subsolutions of the Hamilton-Jacobi equation, Invent. Math. 155 (2004), no. 2, 363-388. MR 2031431 (2004m:37114)
- [Fe69] H. Federer, Geometric Measure Theory, Springer, 1969. MR 0257325 (41:1976)
- [Fe70] H. 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 0260981 (41:5601)
- [FF60] H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458-520. MR 0123260 (23:A588)
- [Fer77] J. Ferrand, Sur la régularité des applications conformes, C.R. Acad. Sci. Paris Ser. A-B 284 (1977), no. 1, A77-A79. MR 0470894 (57:10638)
- [Gr83] M. Gromov, Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1-147. MR 0697984 (85h:53029)
- [Gr96] M. Gromov, Systoles and intersystolic inequalities, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291-362, Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996. www.emis.de/journals/SC/1996/1/ps/smf_sem-cong_1_291-362.ps.gz MR 1427763 (99a:53051)
- [Gr99] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progr. in Mathematics 152, Birkhäuser, Boston, 1999. MR 1699320 (2000d:53065)
- [Ha92] C. Hamburger, Regularity of differential forms minimizing degenerate functionals, J. reine angew. Math. 431 (1992), 7-64. MR 1179331 (93i:49049)
- [Iv02] S. Ivanov, On two-dimensional minimal fillings, St. Petersburg Math. J. 13 (2002), no. 1, 17-25. MR 1819361 (2002b:58016)
- [IK04] S. Ivanov and M. Katz, Generalized degree and optimal Loewner-type inequalities, Israel J. Math. 141 (2004), 221-233. arXiv:math.DG/0405019 MR 2063034
- [Jo48] F. 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, NY, 1948, pp. 187-204. MR 0030135 (10:719b)
- [Ka03] M. Katz, Four-manifold systoles and surjectivity of period map, Comment. Math. Helv. 78 (2003), 772-786. arXiv:math.DG/0302306 MR 2016695 (2005d:53062)
- [Ka06] M. Katz, Systolic geometry and topology, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, to appear.
- [KL04] M. Katz and C. Lescop, Filling area conjecture, optimal systolic inequalities, and the fiber class in abelian covers, Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Math., vol. 387, Amer. Math. Soc., Providence, RI, 2005. See arXiv:math.DG/0412011 MR 2180208 (2006h:53030)
- [KR04] 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
- [KR05] M. Katz and Y. Rudyak, Bounding volume by systoles of 3-manifolds. See arXiv:math.DG/0504008
- [KS04] M. Katz and S. Sabourau, Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1189-1195. Available at the site arXiv:math.DG/0407009 MR 2196056 (2006j:53054)
- [KS05] M. Katz and S. Sabourau, Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory and Dynamical Systems 25 (2005), no. 4, 1209-1220. See arXiv:math.DG/0410312 MR 2158402 (2006d:53038)
- [KS06] 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
- [Lê93] Vân Lê Hông, Curvature estimate for the volume growth of globally minimal submanifolds, Math. Ann. 296 (1993), no. 1, 103-118. MR 1213374 (94a:53095)
- [Li69] A. Lichnerowicz, Applications harmoniques dans un tore, C.R. Acad. Sci., Sér. A 269 (1969), 912-916. MR 0253254 (40:6469)
- [Mo95] F. Morgan, Geometric measure theory. A beginner's guide. Second Edition, Academic Press, Boston, MA, 1995. MR 1326605 (96c:49001)
- [Na04] P.-A. Nagy, On length and product of harmonic forms in Kaehler geometry. See arXiv:math.DG/0406341
- [NV04] P.-A. Nagy and C. Vernicos, The length of harmonic forms on a compact Riemannian manifold, Trans. Amer. Math. Soc. 356 (2004), 2501-2513. Available at arXiv:math.DG/0301369 MR 2048527 (2005f:58055)
- [Pu52] P.M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55-71. MR 0048886 (14:87e)
- [To84] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150. MR 0727034 (85g:35047)
- [Wh99] B. White, The deformation theorem for flat chains, Acta Math. 183 (1999), no. 2, 255-271. MR 1738045 (2000m:49060)
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C23, 57N65, 52C07
Retrieve articles in all journals with MSC (2000): 53C23, 57N65, 52C07
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
Email:
ccroke@math.upenn.edu
Sergei V. Ivanov
Affiliation:
Steklov Mathematics Institute, Fontanka 27, RU-191011 St. Petersburg, Russia
Email:
svivanov@pdmi.ras.ru
Mikhail G. Katz
Affiliation:
Department of Mathematics, Bar Ilan University, Ramat Gan 52900, Israel
Email:
katzmik@math.biu.ac.il
DOI:
https://doi.org/10.1090/S0002-9947-06-03836-0
Keywords:
Abel-Jacobi map,
conformal systole,
deformation theorem,
generalized degree,
extremal lattice,
free abelian cover,
isoperimetric inequality,
John ellipsoid,
$L^p$-minimizing differential forms,
Loewner inequality,
perfect lattice,
Riemannian submersion,
stable systole,
systolic inequality
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)
Article copyright:
© Copyright 2006
American Mathematical Society