Book Review
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Book Information:
Author:
Oliver Knill
Title:
Jürgen Moser, selected chapters in the calculus of variations
Additional book information:
Birkhäuser Verlag,
Basel-Boston-Berlin,
2003,
iv+132 pp.,
ISBN 3-7643-2185-7,
EUR 25.68
Franz Auer and Victor Bangert, Minimising currents and the stable norm in codimension one, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 12, 1095–1100 (English, with English and French summaries). MR 1881240, DOI 10.1016/S0764-4442(01)02188-7
S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D 8 (1983), no. 3, 381–422. MR 719634, DOI 10.1016/0167-2789(83)90233-6
Victor Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 263–286. MR 1062758, DOI 10.1017/S014338570000554X
Victor Bangert, Existence of a complex line in tame almost complex tori, Duke Math. J. 94 (1998), no. 1, 29–40. MR 1635896, DOI 10.1215/S0012-7094-98-09403-0
D. Burago, S. Ivanov, and B. Kleiner, On the structure of the stable norm of periodic metrics, Math. Res. Lett. 4 (1997), no. 6, 791–808. MR 1492121, DOI 10.4310/MRL.1997.v4.n6.a2
Luis A. Caffarelli and Rafael de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math. 54 (2001), no. 12, 1403–1441. MR 1852978, DOI 10.1002/cpa.10008
[Fat] A. Fathi, Weak KAM Theorem and Lagrangian Dynamics, in preparation.
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Gustav A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. (2) 33 (1932), no. 4, 719–739. MR 1503086, DOI 10.2307/1968215
W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Invent. Math. 14 (1971), 63–82 (German). MR 296975, DOI 10.1007/BF01418743
John N. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), no. 4, 457–467. MR 670747, DOI 10.1016/0040-9383(82)90023-4
John N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), no. 2, 207–263. MR 1080112, DOI 10.1090/S0894-0347-1991-1080112-5
John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169–207. MR 1109661, DOI 10.1007/BF02571383
John N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349–1386 (English, with English and French summaries). MR 1275203
[Mat5] J.N. Mather, Arnold diffusion I: Announcement of results, preprint, 2002.
Harold Marston Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924), no. 1, 25–60. MR 1501263, DOI 10.1090/S0002-9947-1924-1501263-9
Jürgen Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 3, 229–272. MR 847308
Jürgen Moser, On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by Jürgen Pöschel), Invent. Math. 119 (1995), no. 3, 401–442. MR 1317647, DOI 10.1007/BF01245189
Paul H. Rabinowitz and Ed Stredulinsky, Mixed states for an Allen-Cahn type equation, Comm. Pure Appl. Math. 56 (2003), no. 8, 1078–1134. Dedicated to the memory of Jürgen K. Moser. MR 1989227, DOI 10.1002/cpa.10087
Zhihong Xia, Arnold diffusion: a variational construction, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 867–877. MR 1648133
- [AB]
- F. Auer and V. Bangert, Minimising currents and the stable norm in codimension one, C.R. Acad. Sci. Paris, Série I, 333 (2001), 1095-1100. MR 1881240 (2002i:49054)
- [AL]
- S. Aubry and P.Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I: Exact results for the ground states, Physica 8D (1983), 381-422. MR 0719634 (85f:58032)
- [Ban1]
- V. Bangert, Minimal geodesics, Erg. Th. Dynam. Sys. 10 (1990), 263-286. MR 1062758 (91j:58126)
- [Ban2]
- V. Bangert, Existence of a complex line in tame almost complex tori, Duke Math. J. 94 (1998), 29-40. MR 1635896 (99j:58030)
- [BIK]
- D. Burago, S. Ivanov and B. Kleiner, On the structure of the stable norm of periodic metrics, Math. Res. Lett. 4 (1997), 791-808. MR 1492121 (98k:53051)
- [CL]
- L.A. Caffarelli and R. de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math. 54 (2001), 1403-1441. MR 1852978 (2002j:49003)
- [Fat]
- A. Fathi, Weak KAM Theorem and Lagrangian Dynamics, in preparation.
- [Gro]
- M. Gromov, Hyperbolic Groups, in: Essays in Group Theory (S. Gersten, ed.), MSRI Publications 8, pp. 75-265, Springer, 1987. MR 0919829 (89e:20070)
- [Hed]
- G.A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. 33 (1932), 719-739. MR 1503086
- [Kli]
- W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Invent. Math. 14 (1971), 63-82. MR 0296975 (45:6034)
- [Mat1]
- J.N. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus, Topology 21 (1982), 465-476. MR 0670747 (84g:58084)
- [Mat2]
- J.N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), 207-263. MR 1080112 (92d:58139)
- [Mat3]
- J.N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), 169-207. MR 1109661 (92m:58048)
- [Mat4]
- J.N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier 43 (1993), 1349-1386. MR 1275203 (95c:58075)
- [Mat5]
- J.N. Mather, Arnold diffusion I: Announcement of results, preprint, 2002.
- [Mor]
- M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Am. Math. Soc. 26 (1924), 25-60. MR 1501263
- [Mos1]
- J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincaré - Anal. Non Linéaire 3 (1986), 229-272. MR 0847308 (88a:58058)
- [Mos2]
- J. Moser, On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by Jürgen Pöschel), Invent. Math. 119 (1995), 401-442. MR 1317647 (96a:58029)
- [RS]
- P. H. Rabinowitz and E. Stredulinsky, Mixed states for an Allen-Cahn type equation, Comm. Pure Appl. Math. 56 (2003), 1078-1134. MR 1989227 (2004i:35125)
- [Xia]
- Z. Xia, Arnold diffusion: a variational construction, Proc. Internat. Congr. Math. 1998, Vol. II, 867-877. MR 1648133 (99g:58112)
Review Information:
Reviewer:
Victor Bangert
Affiliation:
Mathematisches Institut der Universität Freiburg
Email:
bangert@email.mathematik.uni-freiburg.de
Journal:
Bull. Amer. Math. Soc.
42 (2005), 245-249
Published electronically:
January 12, 2005
Review copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.