|
The conformal Yamabe constant of product manifolds
Authors:
Bernd Ammann, Mattias Dahl and Emmanuel Humbert
Journal:
Proc. Amer. Math. Soc. 141 (2013), 295-307
MSC (2010):
Primary 35J60; Secondary 35P30, 58J50, 58C40
Posted:
May 23, 2012
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Let  and  be compact Riemannian manifolds of dimension at least  . We derive a lower bound for the conformal Yamabe constant of the product manifold  in terms of the conformal Yamabe constants of  and  .
- 1.
Kazuo
Akutagawa, Luis
A. Florit, and Jimmy
Petean, On Yamabe constants of Riemannian products, Comm.
Anal. Geom. 15 (2007), no. 5, 947–969. MR 2403191
(2009i:53030)
- 2.
B. Ammann, M. Dahl, and E. Humbert, Smooth Yamabe invariant and surgery, Preprint, arXiv 0804.1418, 2008.
- 3.
-, Low-dimensional surgery and the Yamabe invariant, Preprint in preparation, 2011.
- 4.
-, Square-integrability of solutions of the Yamabe equation, Preprint, arXiv:1111.2780, 2011.
- 5.
Thierry
Aubin, Équations différentielles non linéaires
et problème de Yamabe concernant la courbure scalaire, J. Math.
Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 0431287
(55 #4288)
- 6.
A.
Benedek and R.
Panzone, The space 𝐿^{𝑝}, with mixed norm,
Duke Math. J. 28 (1961), 301–324. MR 0126155
(23 #A3451)
- 7.
L. Bérard Bergery and G. Kaas, Examples of multiple solutions for the Yamabe problem on scalar curvature, Preprint, http://hal.archives-ouvertes.fr/hal-00143495/.
- 8.
-, Remark on an example by R. Schoen concerning the scalar curvature, Preprint, http://hal.archives-ouvertes.fr/hal-00143485/.
- 9.
C.
Böhm, M.
Wang, and W.
Ziller, A variational approach for compact homogeneous Einstein
manifolds, Geom. Funct. Anal. 14 (2004), no. 4,
681–733. MR 2084976
(2005g:53074), http://dx.doi.org/10.1007/s00039-004-0471-x
- 10.
L. L. de Lima, P. Piccione, and M. Zedda, A note on the uniqueness of solutions for the Yamabe problem, to appear in Proc. Amer. Math. Soc., 2011, arXiv:1102.2321.
- 11.
N. Große, The Yamabe equation on manifolds of bounded geometry, Preprint, 2009, arXiv:0912.4398.
- 12.
G. Henry and J. Petean, Isoparametric hypersurfaces and metrics of constant scalar curvature, Preprint, CIMAT Mexico, 2011.
- 13.
Seongtag
Kim, An obstruction to the conformal
compactification of Riemannian manifolds, Proc.
Amer. Math. Soc. 128 (2000), no. 6, 1833-1838. MR 1646195
(2000j:53048), http://dx.doi.org/10.1090/S0002-9939-99-05207-7
- 14.
John
M. Lee and Thomas
H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91. MR 888880
(88f:53001), http://dx.doi.org/10.1090/S0273-0979-1987-15514-5
- 15.
Morio
Obata, The conjectures on conformal transformations of Riemannian
manifolds, J. Differential Geometry 6 (1971/72),
247–258. MR 0303464
(46 #2601)
- 16.
Jimmy
Petean, Best Sobolev constants and manifolds with positive scalar
curvature metrics, Ann. Global Anal. Geom. 20 (2001),
no. 3, 231–242. MR 1866416
(2002h:53064), http://dx.doi.org/10.1023/A:1012037030262
- 17.
Jimmy
Petean, Isoperimetric regions in spherical cones and Yamabe
constants of 𝑀×𝑆¹, Geom. Dedicata
143 (2009), 37–48. MR 2576291
(2011b:53078), http://dx.doi.org/10.1007/s10711-009-9370-5
- 18.
Jimmy
Petean, Metrics of constant scalar curvature
conformal to Riemannian products, Proc. Amer.
Math. Soc. 138 (2010), no. 8, 2897–2905. MR 2644902
(2011j:53061), http://dx.doi.org/10.1090/S0002-9939-10-10293-7
- 19.
Jimmy
Petean and Juan
Miguel Ruiz, Isoperimetric profile comparisons and Yamabe
constants, Ann. Global Anal. Geom. 40 (2011),
no. 2, 177–189. MR 2811624
(2012i:53027), http://dx.doi.org/10.1007/s10455-011-9252-6
- 20.
-, On the Yamabe constants of
 and  , Preprint, arXiv:1202.1022, 2011.
- 21.
Richard
Schoen, Conformal deformation of a Riemannian metric to constant
scalar curvature, J. Differential Geom. 20 (1984),
no. 2, 479–495. MR 788292
(86i:58137)
- 22.
Richard
M. Schoen, On the number of constant scalar curvature metrics in a
conformal class, Differential geometry, Pitman Monogr. Surveys Pure
Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991,
pp. 311–320. MR 1173050
(94e:53035)
- 23.
Neil
S. Trudinger, Remarks concerning the conformal deformation of
Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup.
Pisa (3) 22 (1968), 265–274. MR 0240748
(39 #2093)
- 1.
- K. Akutagawa, L. Florit, and J. Petean, On the Yamabe constant of Riemannian products, Comm. Anal. Geom. 15 (2007), 947-969. MR 2403191 (2009i:53030)
- 2.
- B. Ammann, M. Dahl, and E. Humbert, Smooth Yamabe invariant and surgery, Preprint, arXiv 0804.1418, 2008.
- 3.
- -, Low-dimensional surgery and the Yamabe invariant, Preprint in preparation, 2011.
- 4.
- -, Square-integrability of solutions of the Yamabe equation, Preprint, arXiv:1111.2780, 2011.
- 5.
- T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pure Appl., IX. Ser. 55 (1976), 269-296. MR 0431287 (55:4288)
- 6.
- A. Benedek and R. Panzone, The space
 , with mixed norm, Duke Math. J. 28 (1961), 301-324. MR 0126155 (23:A3451)
- 7.
- L. Bérard Bergery and G. Kaas, Examples of multiple solutions for the Yamabe problem on scalar curvature, Preprint, http://hal.archives-ouvertes.fr/hal-00143495/.
- 8.
- -, Remark on an example by R. Schoen concerning the scalar curvature, Preprint, http://hal.archives-ouvertes.fr/hal-00143485/.
- 9.
- C. Böhm, M. Wang, and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 (2004), 681-733. MR 2084976 (2005g:53074)
- 10.
- L. L. de Lima, P. Piccione, and M. Zedda, A note on the uniqueness of solutions for the Yamabe problem, to appear in Proc. Amer. Math. Soc., 2011, arXiv:1102.2321.
- 11.
- N. Große, The Yamabe equation on manifolds of bounded geometry, Preprint, 2009, arXiv:0912.4398.
- 12.
- G. Henry and J. Petean, Isoparametric hypersurfaces and metrics of constant scalar curvature, Preprint, CIMAT Mexico, 2011.
- 13.
- S. Kim, An obstruction to the conformal compactification of Riemannian manifolds, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1833-1838. MR 1646195 (2000j:53048)
- 14.
- J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc., New Ser. 17 (1987), 37-91. MR 888880 (88f:53001)
- 15.
- M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Diff. Geom. 6 (1971/72), 247-258. MR 0303464 (46:2601)
- 16.
- J. Petean, Best Sobolev constants and manifolds with positive scalar curvature metrics, Ann. Global Anal. Geom. 20 (2001), 231-242. MR 1866416 (2002h:53064)
- 17.
- -, Isoperimetric regions in spherical cones and Yamabe constants of
 , Geom. Dedicata 143 (2009), 37-48. MR 2576291 (2011b:53078)
- 18.
- -, Metrics of constant scalar curvature conformal to Riemannian products, Proc. Amer. Math. Soc. 138 (2010), 2897-2905. MR 2644902
- 19.
- J. Petean and J. M. Ruiz, Isoperimetric profile comparisons and Yamabe constants, Ann. Global Anal. Geom. 40 (2011), 177-189. MR 2811624
- 20.
- -, On the Yamabe constants of and , Preprint, arXiv:1202.1022, 2011.
- 21.
- R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature,
J. Diff. Geom. 20 (1984), 479-495. MR 788292 (86i:58137)
- 22.
- -, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311-320. MR 1173050 (94e:53035)
- 23.
- N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), 265-274. MR 0240748 (39:2093)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
35J60,
35P30,
58J50,
58C40
Retrieve articles in all journals
with MSC (2010):
35J60,
35P30,
58J50,
58C40
Additional Information
Bernd Ammann
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
Email:
bernd.ammann@mathematik.uni-regensburg.de
Mattias Dahl
Affiliation:
Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden
Email:
dahl@math.kth.se
Emmanuel Humbert
Affiliation:
Institut Élie Cartan, BP 239, Université de Nancy 1, 54506 Vandoeuvre-lès-Nancy Cedex, France
Address at time of publication:
Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmot, 37200 Tours, France
Email:
emmanuel.humbert@lmpt.univ-tours.fr
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11320-6
PII:
S 0002-9939(2012)11320-6
Keywords:
Yamabe constant,
Yamabe invariant,
product manifolds
Received by editor(s):
March 9, 2011
Received by editor(s) in revised form:
June 22, 2011
Posted:
May 23, 2012
Additional Notes:
The first author was partially supported by DFG Sachbeihilfe AM 144/2-1.
The second author was partially supported by the Swedish Research Council.
The third author was partially supported by ANR-10-BLAN 0105.
Communicated by:
Michael Wolf
Article copyright:
© Copyright 2012 American Mathematical Society
|
