Abstract
In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.
References
[1] L. Ambrosio and G. Alberti. A geometric approach to monotone functions in Rn. Math. Z., 230: 259–316 (1999). 10.1007/PL00004691Search in Google Scholar
[2] A. D. Aleksandrov and V. A. Zalgaller. Intrinsic geometry of surfaces. Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 15. American Mathematical Society, Providence, R.I., 1967. Search in Google Scholar
[3] A. L. Besse. Einstein manifolds. Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. Search in Google Scholar
[4] A. D. Alexandrov. A. D. Alexandrov selected works. Part II. Chapman & Hall/CRC, Boca Raton, FL, 2006. Intrinsic geometry of convex surfaces, Edited by S. S. Kutateladze, Translated from the Russian by S. Vakhrameyev. Search in Google Scholar
[5] D. Burago, Yu. Burago, and S. Ivanov. A course in metric geometry, volume 33 of Graduate Studies inMathematics. American Mathematical Society, Providence, RI, 2001. 10.1090/gsm/033Search in Google Scholar
[6] Yu. Burago, M. Gromov, and G. Perel0man. A. D. Aleksandrov spaces with curvatures bounded below. Uspekhi Mat. Nauk, 47(2(284)):3–51, 222, 1992. 10.1070/RM1992v047n02ABEH000877Search in Google Scholar
[7] L. Hörmander. The analysis of linear partial differential operators. I. Classics inMathematics. Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. Search in Google Scholar
[8] L. Hörmander. The analysis of linear partial differential operators. II. Classics inMathematics. Springer-Verlag, Berlin, 2005. Differential operators with constant coefficients, Reprint of the 1983 original. 10.1007/b138375Search in Google Scholar
[9] Alfred Huber. Zumpotentialtheoretischen Aspekt der Alexandrowschen Flächentheorie. Comment.Math. Helv., 34:99–126, 1960. 10.1007/BF02565931Search in Google Scholar
[10] Y. Machigashira. The Gaussian curvature of Alexandrov surfaces. J. Math. Soc. Japan, 50(4):859–878, 1998. 10.2969/jmsj/05040859Search in Google Scholar
[11] Y. Otsu and T. Shioya. The Riemannian structure of Alexandrov spaces. J. Differential Geom., 39(3):629–658, 1994. 10.4310/jdg/1214455075Search in Google Scholar
[12] G. Perel0man. DC Structure on Alexandrov Space. Preprint, 1994. Search in Google Scholar
[13] G. Ya. Perel0man. Elements of Morse theory on Aleksandrov spaces. Algebra i Analiz, 5(1):232–241, 1993. Search in Google Scholar
[14] Yu. Reshetnyak. Two-DimensionalManifolds of Bounded Curvature. Geometry. IV, volume 70 of Encyclopaedia ofMathematical Sciences. Springer-Verlag, Berlin, 1993. Nonregular Riemannian geometry, A translation of ıt Geometry, 4 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [ MR1099201 (91k:53003)], Translation by E. Primrose. Search in Google Scholar
[15] Michael Spivak. A comprehensive introduction to differential geometry. Vol. II. Publish or Perish, Inc., Wilmington, Del., second edition, 1979. Search in Google Scholar
[16] M. Troyanov. Un principe de concentration-compacité pour les suites de surfaces riemanniennes. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(5):419–441, 1991. 10.1016/s0294-1449(16)30255-4Search in Google Scholar
[17] M. Troyanov. Les surfaces à courbure intégrale bornée au sens d’Alexandrov. Available on arXiv:0906.3407v1. Search in Google Scholar
© 2016 Luigi Ambrosio and Jérôme Bertrand
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