Non-existence of concave functions on certain metric spaces
HTML articles powered by AMS MathViewer
- by Yin Jiang
- Proc. Amer. Math. Soc. 151 (2023), 2185-2200
- DOI: https://doi.org/10.1090/proc/16287
- Published electronically: February 17, 2023
- HTML | PDF | Request permission
Abstract:
Yau [Math. Ann. 207 (1974), pp. 269–270] proved that: There is no non-trivial continuous concave function on a complete manifold with finite volume. We prove analogue theorems for several metric spaces, including Alexandrov spaces with curvature bounded below/above, $C^{\alpha }$-Hölder Riemannian manifolds.References
- S. Alexander, V. Kapovitch, and A. Petrunin, Alexandrov geometry, arXiv:1903.08539, 2019.
- Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin, An invitation to Alexandrov geometry, SpringerBriefs in Mathematics, Springer, Cham, 2019. CAT(0) spaces. MR 3930625, DOI 10.1007/978-3-030-05312-3
- Luigi Ambrosio and Bernd Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000), no. 3, 527–555. MR 1800768, DOI 10.1007/s002080000122
- R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
- Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
- Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
- Karsten Grove and Peter Petersen, A radius sphere theorem, Invent. Math. 112 (1993), no. 3, 577–583. MR 1218324, DOI 10.1007/BF01232447
- Jürgen Jost, Nonlinear Dirichlet forms, New directions in Dirichlet forms, AMS/IP Stud. Adv. Math., vol. 8, Amer. Math. Soc., Providence, RI, 1998, pp. 1–47. MR 1652278, DOI 10.1090/amsip/008/01
- Vitali Kapovitch, Alexander Lytchak, and Anton Petrunin, Metric-measure boundary and geodesic flow on Alexandrov spaces, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 1, 29–62. MR 4186463, DOI 10.4171/jems/1006
- A. Lytchak, Differentiation in metric spaces, Algebra i Analiz 16 (2004), no. 6, 128–161; English transl., St. Petersburg Math. J. 16 (2005), no. 6, 1017–1041. MR 2117451, DOI 10.1090/S1061-0022-05-00888-5
- A. Lytchak, Open map theorem for metric spaces, Algebra i Analiz 17 (2005), no. 3, 139–159; English transl., St. Petersburg Math. J. 17 (2006), no. 3, 477–491. MR 2167848, DOI 10.1090/S1061-0022-06-00916-2
- Alexander Lytchak and Asli Yaman, On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 (2006), no. 7, 2917–2926. MR 2216252, DOI 10.1090/S0002-9947-06-04195-X
- Alexander Lytchak and Koichi Nagano, Geodesically complete spaces with an upper curvature bound, Geom. Funct. Anal. 29 (2019), no. 1, 295–342. MR 3925112, DOI 10.1007/s00039-019-00483-7
- Uwe F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), no. 2, 199–253. MR 1651416, DOI 10.4310/CAG.1998.v6.n2.a1
- Anton Petrunin, Semiconcave functions in Alexandrov’s geometry, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137–201. MR 2408266, DOI 10.4310/SDG.2006.v11.n1.a6
- G. Perelman and A. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, eprint: http://www.math.psu.edu/petrunin/.
- R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876
- V. A. Šarafutdinov, The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to $\textbf {R}^{n}$, Sibirsk. Mat. Ž. 18 (1977), no. 4, 915–925, 958 (Russian). MR 487896
- Katsuhiro Shiohama, Complete noncompact Alexandrov spaces of nonnegative curvature, Arch. Math. (Basel) 60 (1993), no. 3, 283–289. MR 1201643, DOI 10.1007/BF01198813
- Shing Tung Yau, Non-existence of continuous convex functions on certain Riemannian manifolds, Math. Ann. 207 (1974), 269–270. MR 339008, DOI 10.1007/BF01351342
- H. C. Zhang, note of FDSCBB, unpublished notes.
Bibliographic Information
- Yin Jiang
- Affiliation: School of Mathematical Sciences, Beihang University, Beijing People’s Republic of China
- Email: jiangyin@buaa.edu.cn
- Received by editor(s): July 25, 2022
- Received by editor(s) in revised form: August 4, 2022
- Published electronically: February 17, 2023
- Additional Notes: The author was partially supported by NSFC 11901023.
- Communicated by: Lu Wang
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2185-2200
- MSC (2020): Primary 53C23, 51F30, 53C45
- DOI: https://doi.org/10.1090/proc/16287
- MathSciNet review: 4556210