Previous |  Up |  Next

Article

Title: Transversal biwave maps (English)
Author: Chiang, Yuan-Jen
Author: Wolak, Robert A.
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 46
Issue: 3
Year: 2010
Pages: 211-226
Summary lang: English
.
Category: math
.
Summary: In this paper, we prove that the composition of a transversal biwave map and a transversally totally geodesic map is a transversal biwave map. We show that there are biwave maps which are not transversal biwave maps, and there are transversal biwave maps which are not biwave maps either. We prove that if $f$ is a transversal biwave map satisfying certain condition, then $f$ is a transversal wave map. We finally study the transversal conservation laws of transversal biwave maps. (English)
Keyword: transversal bi-energy
Keyword: transversal biwave field
Keyword: transversal biwave map
MSC: 53C12
MSC: 58E20
MSC: 58G11
idZBL: Zbl 1240.58009
idMR: MR2735907
.
Date available: 2010-10-22T05:37:57Z
Last updated: 2013-09-19
Stable URL: http://hdl.handle.net/10338.dmlcz/140789
.
Reference: [1] Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds in spheres.Internat. J. Math. 12 (8) (2001), 867–876. MR 1863283, 10.1142/S0129167X01001027
Reference: [2] Caddeo, R., Montaldo, S., Piu, P.: Explicit formulas for non-geodesic biharmonic curves of the Heisenberg group.Rend. Sem. Mat. Univ. Politec. Torino 62 (3) (2004), 265–278. MR 2129448
Reference: [3] Chang, Sun-Yung A., Wang, L., Yang, P. C.: Regularity of biharmonic maps.Comm. Pure Appl. Math. 52 (9) (1999), 1099–1111. MR 1692152, 10.1002/(SICI)1097-0312(199909)52:9<1099::AID-CPA3>3.0.CO;2-O
Reference: [4] Chiang, Y. J.: Biwave maps into manifolds.Internat. J. Math. Math. Sci. Article ID 104274 (2009), 1–14. Zbl 1169.53336, MR 2515931, 10.1155/2009/104274
Reference: [5] Chiang, Y. J., Sun, H.: 2-harmonic totally real submanifolds in a complex projective space.Bull. Inst. Math. Acad. Sinica 27 (2) (1999), 99–107. Zbl 0960.53036, MR 1697219
Reference: [6] Chiang, Y. J., Sun, H.: Biharmonic Maps on V-Manifolds.Internat. J. Math. Math. Sci. 27 (2001), 477–484. Zbl 1012.58012, MR 1869649, 10.1155/S0161171201006731
Reference: [7] Chiang, Y. J., Wolak, R.: Transversally biharmonic maps between foliated Riemannian manifolds.Internat. J. Math. 19 (9) (2008), 1–16. Zbl 1160.53012, MR 2446510, 10.1142/S0129167X08004972
Reference: [8] Chiang, Y. J., Yang, Y. H.: Exponential wave maps.J. Geom. Phys. 57 (2007), 2521–2532. Zbl 1134.58006, MR 2369837, 10.1016/j.geomphys.2007.09.003
Reference: [9] Eells, J., Lemaire, L.: A report on harmonic maps.Bull. London Math. Soc. 10 (1978), 1–68. Zbl 0401.58003, MR 0495450, 10.1112/blms/10.1.1
Reference: [10] Eells, J., Lemaire, L.: Another report on harmonic maps.Bull. London Math. Soc. 20 (1988), 385–524. Zbl 0669.58009, MR 0956352, 10.1112/blms/20.5.385
Reference: [11] Eells, J., Sampson, J. H.: Harmonic mappings between Riemannian manifolds.Amer. J. Math. 86 (1964), 109–160. MR 0164306, 10.2307/2373037
Reference: [12] Eells, J., Verjovsky, A.: Harmonic and Riemannian foliations.Bol. Soc. Mat. Mexicana (3) 4 (1998), 1–12. Zbl 0905.57019, MR 1625589
Reference: [13] Evans, L. C.: Partial Differential Equations.vol. 19, Grad. Stud. Math., 1998. Zbl 0902.35002
Reference: [14] Hilbert, D.: Die Grundlagen der Physik.Math. Ann. 92 (1924), 1–32. MR 1512197, 10.1007/BF01448427
Reference: [15] Jiang, G. Y.: 2-harmonic maps between Riemannian manifolds.Annals of Math., China 7A (4) (1986), 389–402. MR 0886529
Reference: [16] Jiang, G. Y.: The 2-harmonic isometric immersions between Riemannian manifolds.Annals of Math., China 7A (2) (1986), 130–144. MR 0858581
Reference: [17] Jiang, G. Y.: the conservation law of 2-harmonic maps between Riemannian manifolds.Acta Math. Sinica 30 (2) (1987), 220–225. MR 0891928
Reference: [18] Kacimi, A. El, Gomez, E. Gallego: Foliated harmonic maps.Illinois J. Math. 40 (1996), 115–122. MR 1386316
Reference: [19] Klainerman, S., Machedon, M.: Smoothing estimates for null forms and applications.Duke Math. J. 81 (1995), 99–133. Zbl 0909.35094, MR 1381973, 10.1215/S0012-7094-95-08109-5
Reference: [20] Klainerman, S., Machedon, M.: On the optimal local regularity for gauge fields theories.Differential Integral Equations 10 (6) (1997), 1019–1030. MR 1608017
Reference: [21] Konderak, J. J., Wolak, R.: Transversally harmonic maps between manifolds with Riemannian foliations.Quart. J. Math. Oxford Ser. (2) 54 (Part 3) (2003), 335–354. Zbl 1059.53051, MR 2013142, 10.1093/qmath/hag019
Reference: [22] Konderak, J. J., Wolak, R.: Some remarks on transversally harmonic maps.Glasgow J. Math. 50 (1) (2008), 1–16. Zbl 1138.53025, MR 2381726
Reference: [23] Lopez, J. A., Masa, X. M.: Morphisms between complete Riemannian pseudogroups.Topology Appl. 155 (2008), 544–604. Zbl 1147.57025, MR 2388956, 10.1016/j.topol.2007.12.001
Reference: [24] Loubeau, E., Oniciuc, C.: On the biharmonic and harmonic indices of the Hopf map.Trans. Amer. Math. Soc. 359 (11) (2007), 5239–5256. Zbl 1124.58009, MR 2327029, 10.1090/S0002-9947-07-03934-7
Reference: [25] Molino, P.: Riemannian Foliations.Birkhauser, Bassel, 1988. Zbl 0824.53028, MR 0932463
Reference: [26] Montaldo, S., Oniciuc, C.: A short survey on biharmonic maps between Riemannian manifolds.Rev. Un. Mat. Argentina 47 (2006), 1–22. Zbl 1140.58004, MR 2301373
Reference: [27] Nahmod, A., Stefanov, A., Uhlenbeck, K.: On the well-posedness of the wave map problem in high dimensions.Comm. Anal. Geom. 11 (1) (2003), 49–83. Zbl 1085.58022, MR 2016196
Reference: [28] O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity.Academic Press, 1983. MR 0719023
Reference: [29] Shatah, J., Struwe, M.: Geometric wave equations.Courant Lecture Notes in Math., 2 (2000), viii+153 pp. Zbl 1051.35500, MR 1674843
Reference: [30] Shatah, J., Struwe, M.: The Cauchy problem for wave maps.Int. Math. Res. Not. 11 (2002), 555–571. Zbl 1024.58014, MR 1890048, 10.1155/S1073792802109044
Reference: [31] Tao, T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension.Internat. Math. Res. Notices 6 (2001), 299–328. Zbl 0983.35080, MR 1820329
Reference: [32] Tao, T.: Global regularity of wave maps. II. Small energy in two dimension.Comm. Math. Phys. 224 (2001), 443–544. MR 1869874, 10.1007/PL00005588
Reference: [33] Tataru, D.: The wave maps equations.Bull. Amer. Math. Soc. 41 (2004), 185–204. MR 2043751, 10.1090/S0273-0979-04-01005-5
Reference: [34] Tataru, D.: Rough solutions for the wave maps equation.Amer. J. Math. 127 (5) (2005), 293–377. MR 2130618, 10.1353/ajm.2005.0014
Reference: [35] Tondeur, P.: Geometry of Foliation.Birkhauser, Bassel, 1997. MR 1456994
Reference: [36] Wang, C.: Biharmonic maps from $R^4$in to a Riemannian manifold.Math. Z. 247 (2004), 65–87. MR 2054520, 10.1007/s00209-003-0620-1
Reference: [37] Wolak, R.: Foliated and associated geometric structures on foliated manifolds.Ann. Fac. Sci. Toulouse Math. (6) 10 (1989), 337–360. Zbl 0698.57007, MR 1425491, 10.5802/afst.681
.

Files

Files Size Format View
ArchMathRetro_046-2010-3_5.pdf 550.4Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo