Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The weighted Sobolev and mean value inequalities


Author: Adriano Alves de Medeiros
Journal: Proc. Amer. Math. Soc. 143 (2015), 1229-1239
MSC (2010): Primary 53C42; Secondary 53C21
DOI: https://doi.org/10.1090/S0002-9939-2014-12337-9
Published electronically: November 24, 2014
MathSciNet review: 3293738
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a Michael-Simon inequality in the weighted setting and using this inequality we obtain a diameter control depending of the $ f$-mean curvature, which is based in the work of Topping.


References [Enhancements On Off] (What's this?)

  • [1] William K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417-491. MR 0307015 (46 #6136)
  • [2] E. Bombieri, E. De Giorgi, and M. Miranda, Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, Arch. Rational Mech. Anal. 32 (1969), 255-267 (Italian). MR 0248647 (40 #1898)
  • [3] Tobias Holck Colding and William P. Minicozzi II, A course in minimal surfaces, Graduate Studies in Mathematics, vol. 121, American Mathematical Society, Providence, RI, 2011. MR 2780140
  • [4] Tobias H. Colding and William P. Minicozzi II, Generic mean curvature flow I: generic singularities, Ann. of Math. (2) 175 (2012), no. 2, 755-833. MR 2993752, https://doi.org/10.4007/annals.2012.175.2.7
  • [5] David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715-727. MR 0365424 (51 #1676)
  • [6] D. Impera and M. Rimoldi, Stability properties and topology at infinity of f-minimal hypersurfaces. arXiv:1302.6160 [math.DG].
  • [7] Jia-Yong Wu and Yu Zheng, Relating diameter and mean curvature for Riemannian submanifolds, Proc. Amer. Math. Soc. 139 (2011), no. 11, 4097-4104. MR 2823054 (2012e:53120), https://doi.org/10.1090/S0002-9939-2011-10848-7
  • [8] V. Maz'ya, (Editor) Sobolev spaces in Mathematics I, II, II.
  • [9] J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $ R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361-379. MR 0344978 (49 #9717)
  • [10] Mario Miranda, Diseguaglianze di Sobolev sulle ipersuperfici minimali, Rend. Sem. Mat. Univ. Padova 38 (1967), 69-79 (Italian). MR 0221350 (36 #4402)
  • [11] M. Batista and H. Mirandola, A Sobolev-type inequality for submanifolds in weighted Riemannian Manifolds, arXiv:1304.2271 [math.DG].
  • [12] Nam Q. Le, Blow up of subcritical quantities at the first singular time of the mean curvature flow, Geom. Dedicata 151 (2011), 361-371. MR 2780756 (2012m:53142), https://doi.org/10.1007/s10711-010-9538-z
  • [13] S. L. Soboleve, On a theorem of functional analysis (Russian). Mat. Sb. 4, no.3 (1938), 471-496. English trans.: Eleven papers in Analysis. Am. Math. Transl. (2) 34, (1963), 39-68.
  • [14] Gu-Ji Tian and Xu-Jia Wang, A class of Sobolev type inequalities, Methods Appl. Anal. 15 (2008), no. 2, 263-276. MR 2481683 (2010e:35016), https://doi.org/10.4310/MAA.2008.v15.n2.a10
  • [15] Peter Topping, Diameter control under Ricci flow, Comm. Anal. Geom. 13 (2005), no. 5, 1039-1055. MR 2216151 (2006m:53101)
  • [16] Peter Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539-546. MR 2410779 (2009b:53100), https://doi.org/10.4171/CMH/135
  • [17] Qiaoling Wang, Complete submanifolds in manifolds of partially non-negative curvature, Ann. Global Anal. Geom. 37 (2010), no. 2, 113-124. MR 2578260 (2011a:53115), https://doi.org/10.1007/s10455-009-9176-6
  • [18] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685 (91e:46046)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C42, 53C21

Retrieve articles in all journals with MSC (2010): 53C42, 53C21


Additional Information

Adriano Alves de Medeiros
Affiliation: Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa, PB, Brazil
Email: adrianoalves@mat.ufpb.br

DOI: https://doi.org/10.1090/S0002-9939-2014-12337-9
Received by editor(s): April 17, 2013
Received by editor(s) in revised form: May 24, 2013
Published electronically: November 24, 2014
Additional Notes: The author would like to thank Gregorio Pacelli Bessa and Jorge Herbert Soares de Lira for stimulating conversations about this subject.
Dedicated: Dedicated to my son João Gabriel
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society