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Generalized degree and optimal Loewner-type inequalities. (English) Zbl 1067.53031
The results presented in the reviewed paper are generalizations of optimal inequalities of C. Loewner and M. Gromov. The unpublished Loewner inequality (circa 1949) relates the length of the shortest noncontractible loop of an arbitrary Riemannian metric on the 2-torus to its surface area. Half a century later M. Gromov obtained an optimal generalization of Loewner’s inequality for a compact manifold of equal dimension and first Betti number.
The authors of this paper prove lower bounds for the total volume in terms of the homotopy systole and the stable systole. Their main tool is the construction of an area-decreasing map to the Jacobi torus. It turns out that one can succesfully combine this construction with the coarea formula, yielding new optimal inequalities. This paper includes also interesting historical remarks and a discussion of the related systolic literature.

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
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[1] I. Babenko and M. Katz,Systolic freedom of orientable manifolds, Annales Scientifiques de l’École Normale Supérieure (Paris)31 (1998), 787–809. · Zbl 0944.53019
[2] V. Bangert, C. Croke, S. Ivanov and M. Katz,Boundary case of equality in optimal Loewner-type inequalities and the filling area conjecture, in preparation. · Zbl 1122.53020
[3] V. Bangert and M. Katz,Stable systolic inequalities and cohomology products, Communications on Pure and Applied Mathematics56 (2003), 979–997. · Zbl 1038.53031
[4] V. Bangert and M. Katz,An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm, arXiv:math.DG/0304494. · Zbl 1068.53027
[5] E. S. Barnes,On a theorem of Voronoi, Proceedings of the Cambridge Philosophical Society53 (1957), 537–539. · Zbl 0078.03703
[6] D. Burago and S. Ivanov,Riemannian tori without conjugate points are flat, Geometric and Functional Analysis4 (1994), 259–269. · Zbl 0808.53038
[7] D. Burago and S. Ivanov,On asymptotic volume of tori, Geometric and Functional Analysis5 (1995), 800–808. · Zbl 0846.53043
[8] I. Chavel,Riemannian Geometry–A Modern Introduction, Cambridge Tracts in Mathematics,108, Cambridge University Press, Cambridge, 1993. · Zbl 0810.53001
[9] C. Croke and M. Katz,Universal volume bounds in Riemannian manifolds, Surveys in Differential Geometry 8, 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.), International Press, Somerville, MA, 2003, pp. 109–137.
[10] H. M. Farkas and I. Kra,Riemann Surfaces, Second edition, Graduate Texts in Mathematics71, Springer-Verlag, New York, 1992.
[11] H. Federer,Geometric Measure Theory, Springer, Berlin, 1969. · Zbl 0176.00801
[12] M. Gromov,Structures métriques pour les variétés riemanniennes (J. Lafontaine and P. Pansu, eds.), Textes Mathématiques, 1, CEDIC, Pairs, 1981.
[13] M. Gromov,Filling Riemannian manifolds, Journal of Differential Geometry18 (1983), 1–147. · Zbl 0515.53037
[14] M. Gromov,Systoles and intersystolic inequalities, inActes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Séminaires et Congrès, Vol. 1, Société Mathématique de France, Paris, 1996, pp. 291–362.
[15] M. Gromov,Metric structures for Riemannian and non-Riemannian spaces, Progress in Mathematics, Vol. 152, Birkhäuser, Boston, 1999. · Zbl 0953.53002
[16] S. Ivanov,On two-dimensional minimal fillings, St. Petersburg Mathematical Journal13 (2002), 17–25. · Zbl 0995.58010
[17] M. Katz,Four-manifold systoles and surjectivity of period map, Commentarii Mathematici Helvetici78 (2003), 772–876. · Zbl 1048.53031
[18] M. Katz, M. Kreck and A. Suciu,Free abelian covers, short loops, stable length, and systolic inequalities, · Zbl 1134.53019
[19] J. C. Lagarias, H. W. Lenstra Jr. and C. P. Schnorr,Bounds for Korkin-Zolotarev reduced bases and successive minima of a lattice and its reciprocal lattice, Combinatorica10 (1990), 343–358. · Zbl 0723.11029
[20] A. Lichnerowicz,Applications harmoniques dans un tore, Comptes Rendus de l’Académie des Sciences, Paris, Série I269 (1969), 912–916. · Zbl 0184.25002
[21] V. D. Milman and G. Schechtman,Asymptotic theory of finite-dimensional normed spaces (with an appendix by M. Gromov), Lecture Notes in Mathematics1200, Springer-Verlag, Berlin, 1986. · Zbl 0606.46013
[22] A. Nabutovsky and R. Rotman,The length of the shortest closed geodesic on a 2-dimansional sphere, International Mathematics Research Notices 2002:3 (2002), 1211–1222. · Zbl 1003.53030
[23] P. M. Pu,Some inequalities in certain nonorientable Riemannian manifolds, Pacific Journal of Mathematics2 (1952), 55–71. · Zbl 0046.39902
[24] S. Sabourau,Filling radius and short closed geodesics of the two-sphere, Bulletin de la Société Mathématique de France, to appear.
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