DOI QR코드

DOI QR Code

OMORI-YAU MAXIMUM PRINCIPLE ON ALEXANDROV SPACES

  • Lee, Hanjin (Global Leadership School Handong Global University, School of Mathematics Korea Institute for Advanced Study)
  • 투고 : 2015.02.27
  • 발행 : 2016.05.01

초록

We prove an Omori-Yau maximum principle on Alexandrov spaces which do not have Perelman singular points and satisfy the infinitesimal Bishop-Gromov condition.

키워드

참고문헌

  1. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000.
  2. L. Ambrosio and B. Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527-555. https://doi.org/10.1007/s002080000122
  3. D. Burago, Y. Burago, and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, vol.33, American Mathematical Society, Providence, RI, 2001.
  4. Y. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curvature bounded below, Russian Math. Surveys 47 (1992), no. 2, 1-58. https://doi.org/10.1070/RM1992v047n02ABEH000877
  5. K. Kuwae, Y. Machigahira, and T. Shioya, Sobolev sapces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238 (2001), no. 2, 269-316. https://doi.org/10.1007/s002090100252
  6. K. Kuwae and T. Shioya, Laplacian comparison for Alexandrov spaces, arXiv:0709.0788, 2007.
  7. K. Kuwae and T. Shioya, Infinitesimal Bishop-Gromov condition for Alexandrov spaces, Probabilistic approach to geometry, 293-302, Adv. Stud. Pure Math., 57, Math. Soc. Japan, Tokyo, 2010.
  8. K. Kuwae and T. Shioya, A topological splitting theorem for weighted Alexandrov spaces, Tohoku Math. J. (2) 63 (2011), no. 1, 59-76. https://doi.org/10.2748/tmj/1303219936
  9. S. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), no. 4, 805-828.
  10. H. Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205-214. https://doi.org/10.2969/jmsj/01920205
  11. Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994), no. 3, 629-658. https://doi.org/10.4310/jdg/1214455075
  12. G. Perelman, DC-structure on Alexandrov spaces, preprint.
  13. A. Petrunin, Alexandrov meets Lott-Villani-Sturm, Munster J. Math. 4 (2011), 53-64.
  14. S. Pigola, M. Rigoli, and A. G. Setti, A remark on the maximum principle and stochastic completeness, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1283-1288. https://doi.org/10.1090/S0002-9939-02-06672-8
  15. J. Rataj and L. Zajicek, Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces, Houston J. Math. 38 (2012), no. 2, 445-467.
  16. A. Ratto, M. Rigoli, and A. G. Setti, On the Omori-Yau maximum principle and its application to differential equations and geometry, J. Funct. Anal. 134 (1995), no. 2, 486-510. https://doi.org/10.1006/jfan.1995.1154
  17. M.-K. von Renesse, Heat kernel comparison on Alexandrov spaces with curvature bounded below, Potential Anal. 21 (2004), no. 2, 151-176. https://doi.org/10.1023/B:POTA.0000025376.45065.80
  18. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. https://doi.org/10.1002/cpa.3160280203
  19. S. T. Yau, A general Schwarz lemma for Kahler manifolds, Amer. J. Math. 100 (1978), no. 1, 197-203. https://doi.org/10.2307/2373880
  20. H. Zhang and X. Zhu, Ricci curvature on Alexandrov spaces and rigidity theorems, Comm. Anal. Geom. 18 (2010), no. 3, 503-553. https://doi.org/10.4310/CAG.2010.v18.n3.a4