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2013 | 11 | 6 | 1039-1055
Tytuł artykułu

Deforming metrics of foliations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
6
Strony
1039-1055
Opis fizyczny
Daty
wydano
2013-06-01
online
2013-03-28
Twórcy
autor
  • Faculty of Mathematics and Computer Science, Institute of Mathematics of Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland, robert.wolak@im.uj.edu.pl
Bibliografia
  • [1] Álvarez López J., Kordyukov Yu.A., Long time behavior of leafwise heat flow for Riemannian foliations, Compositio Math., 2001, 125(2), 129–153 http://dx.doi.org/10.1023/A:1002492700960[Crossref]
  • [2] Brendle S., Ricci Flow and the Sphere Theorem, Grad. Stud. Math., 111, American Mathematical Society, Providence, 2010
  • [3] Candel A., Conlon L., Foliations, I, II, Grad. Stud. Math., 23, 60, American Mathematical Society, Providence, 2000, 2003
  • [4] Czarnecki M., Walczak P., Extrinsic geometry of foliations, In: Foliations 2005, World Scientific, Hackensack, 2006, 149–167 http://dx.doi.org/10.1142/9789812772640_0008[Crossref]
  • [5] Gil-Medrano O., Geometric properties of some classes of Riemannian almost-product manifolds, Rend. Circ. Mat. Palermo, 1983, 32(3), 315–329 http://dx.doi.org/10.1007/BF02848536[Crossref]
  • [6] Jost J., Riemannian Geometry and Geometric Analysis, 6th ed., Universitext, Springer, Heidelberg, 2011 http://dx.doi.org/10.1007/978-3-642-21298-7[Crossref]
  • [7] Lovric M., Min-Oo M., Ruh E.A., Deforming transverse Riemannian metrics of foliations, Asian J. Math., 2000, 4(2), 303–314
  • [8] Milgram A.N., Rosenbloom P.C., Harmonic forms and heat conduction. I. Closed Riemannian manifolds, Proc. Nat. Acad. Sci., 1951, 37, 180–184 http://dx.doi.org/10.1073/pnas.37.3.180[Crossref]
  • [9] Miquel V., Some examples of Riemannian almost-product manifolds, Pacific J. Math., 1984, 111(1), 163–178 http://dx.doi.org/10.2140/pjm.1984.111.163[Crossref]
  • [10] Montesinos A., On certain classes of almost product structures, Michigan Math. J., 1983, 30(1), 31–36 http://dx.doi.org/10.1307/mmj/1029002785[Crossref]
  • [11] Naveira A.M., A classification of Riemannian almost-product manifolds, Rend. Mat., 1983, 3(3), 577–592
  • [12] Nishikawa S., Ramachandran M., Tondeur Ph., The heat equation for Riemannian foliations, Trans. Amer. Math. Soc., 1990, 319(2), 619–630 http://dx.doi.org/10.1090/S0002-9947-1990-0987165-6[Crossref]
  • [13] Oshikiri G., Mean curvature functions of codimension-one foliations, Comment. Math. Helv., 1990, 65(1), 79–84 http://dx.doi.org/10.1007/BF02566594[Crossref]
  • [14] Oshikiri G., Mean curvature functions of codimension-one foliations II, Comment. Math. Helv., 1991, 66(4), 512–520 http://dx.doi.org/10.1007/BF02566662[Crossref]
  • [15] Oshikiri G., A characterization of the mean curvature functions of codimension-one foliations, Tôhoku Math. J., 1997, 49(4), 557–563 http://dx.doi.org/10.2748/tmj/1178225061[Crossref]
  • [16] Oshikiri G., Some properties of mean curvature vectors for codimension-one foliations. Illinois J. Math., 2005, 49(1), 159–166
  • [17] Ponge R., Reckziegel H., Twisted products in pseudo-Riemannian geometry, Geom. Dedicata, 1993, 48(1), 15–25 http://dx.doi.org/10.1007/BF01265674[Crossref]
  • [18] Rovenski V., Integral formulae for a Riemannian manifold with two orthogonal distributions, Cent. Eur. J. Math., 2011, 9(3), 558–577 http://dx.doi.org/10.2478/s11533-011-0026-y[WoS][Crossref]
  • [19] Rovenski V., Walczak P., Topics in Extrinsic Geometry of Codimension-One Foliations, Springer Briefs Math., Springer, New York, 2011 http://dx.doi.org/10.1007/978-1-4419-9908-5[Crossref]
  • [20] Rovenski V., Walczak P.G., Integral formulae on foliated symmetric spaces, Math. Ann., 2012, 352(1), 223–237 http://dx.doi.org/10.1007/s00208-011-0637-4[WoS][Crossref]
  • [21] Royo Prieto J.I., Saralegi-Aranguren M., Wolak R., Cohomological tautness for Riemannian foliations, Russ. J. Math. Phys., 2009, 16(3), 450–466 http://dx.doi.org/10.1134/S1061920809030133[Crossref]
  • [22] Schweitzer P., Walczak P.G., Prescribing mean curvature vectors for foliations, Illinois J. Math., 2004, 48(1), 21–35
  • [23] Sullivan D., Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 1976, 36, 225–255 http://dx.doi.org/10.1007/BF01390011[Crossref]
  • [24] Topping P., Lectures on the Ricci Flow, London Math. Soc. Lecture Note Ser., 325, Cambridge University Press, Cambridge, 2006 http://dx.doi.org/10.1017/CBO9780511721465[Crossref]
  • [25] Walczak P.G., Mean curvature functions for codimension one foliations with all leaves compact, Czechoslovak Math. J., 1984, 34(109)(1), 146–155
  • [26] Walczak P., An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math., 1990, 58(2), 243–252
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0231-y
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