Growth of weighted volume and some applications

Mirjana Milijević; Luis P. Yapu

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 1, page 1-10
  • ISSN: 0044-8753

Abstract

top
We define cut-off functions in order to allow higher growth for Dirichlet energy. Our results are generalizations of the classical Cheng-Yau’s growth conditions of parabolicity. Finally we give some applications to the function theory of Kähler and quaternionic-Kähler manifolds.

How to cite

top

Milijević, Mirjana, and Yapu, Luis P.. "Growth of weighted volume and some applications." Archivum Mathematicum 056.1 (2020): 1-10. <http://eudml.org/doc/295084>.

@article{Milijević2020,
abstract = {We define cut-off functions in order to allow higher growth for Dirichlet energy. Our results are generalizations of the classical Cheng-Yau’s growth conditions of parabolicity. Finally we give some applications to the function theory of Kähler and quaternionic-Kähler manifolds.},
author = {Milijević, Mirjana, Yapu, Luis P.},
journal = {Archivum Mathematicum},
keywords = {volume growth; parabolic manifolds; weighted parabolic manifolds},
language = {eng},
number = {1},
pages = {1-10},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Growth of weighted volume and some applications},
url = {http://eudml.org/doc/295084},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Milijević, Mirjana
AU - Yapu, Luis P.
TI - Growth of weighted volume and some applications
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 1
SP - 1
EP - 10
AB - We define cut-off functions in order to allow higher growth for Dirichlet energy. Our results are generalizations of the classical Cheng-Yau’s growth conditions of parabolicity. Finally we give some applications to the function theory of Kähler and quaternionic-Kähler manifolds.
LA - eng
KW - volume growth; parabolic manifolds; weighted parabolic manifolds
UR - http://eudml.org/doc/295084
ER -

References

top
  1. Cheng, S.Y., Yau, S.T., 10.1002/cpa.3160280303, Comm. Pure Appl. Math. 28 (3) (1975), 333–354. (1975) Zbl0312.53031MR0385749DOI10.1002/cpa.3160280303
  2. Corlette, K., 10.2307/2946567, Ann. of Math. (2) 135 (1) (1992), 165–182. (1992) MR1147961DOI10.2307/2946567
  3. Greene, R.E., Wu, H., 10.1007/BFb0063413, Lecture Notes in Math., vol. 699, Springer-Verlag, Berlin and New York, 1979. (1979) MR0521983DOI10.1007/BFb0063413
  4. Grigoryan, A.A., On the existence of a Green function on a manifold, Uspekhi Mat. Nauk 38 (1983), 161–162, (Russian), English translation: Russian Math. Surveys 38 (1983), no. 1, 190–191. (1983) MR0693728
  5. Grigoryan, A.A., 10.1090/S0273-0979-99-00776-4, Bull. Amer. Math. Soc. 36 (2) (1999), 135–249. (1999) MR1659871DOI10.1090/S0273-0979-99-00776-4
  6. Hua, B., Liu, S., Xia, C., 10.2140/pjm.2017.290.381, Pacific J. Math. 290 (2017), 381–402. (2017) MR3681112DOI10.2140/pjm.2017.290.381
  7. Jost, J., Riemannian geometry and geometric analysis, Springer, Berlin, 2017. (2017) MR3726907
  8. Karp, L., Subharmonic functions, harmonic mappings and isometric immersions, Seminar on Differential Geometry (Yau, S.T., ed.), Ann. Math. Stud. 102, Princeton, 1982. (1982) 
  9. Kong, S., Li, P., Zhou, D., 10.4310/jdg/1203000269, J. Differential Geom. 78 92) (2008), 295–332. (2008) MR2394025DOI10.4310/jdg/1203000269
  10. Lam, K-H., 10.4310/MRL.2008.v15.n6.a8, Math. Res. Lett. 15 (6) (2018), 1167–1186. (2018) MR2470392DOI10.4310/MRL.2008.v15.n6.a8
  11. Li, P., 10.1007/BF01234432, Invent. Math. 99 (1990), 579–600. (1990) MR1032881DOI10.1007/BF01234432
  12. Li, P., Curvature and function theory on Riemannian manifolds, Surveys in Differential Geometry, vol. 7, International Press, Cambridge, 2002, Papers dedicated to Atiyah, Bott, Hirzebruch, and Singer, pp. 375–432. (2002) 
  13. Munteanu, O., Wang, J., 10.1007/s00208-015-1192-1, Math. Ann. 363 (2015), 893–911. (2015) MR3412346DOI10.1007/s00208-015-1192-1
  14. Rigolli, M., Setti, A., 10.4171/RMI/302, Rev. Mat. Iberoamerican 17 (2001), 471–521. (2001) MR1900893DOI10.4171/RMI/302
  15. Ruppenthal, J., 10.1090/proc12718, Proc. Amer. Math. Soc. 144 (2016), 225–233. (2016) MR3415591DOI10.1090/proc12718
  16. Sampson, J.H., Applications to harmonic maps to Kähler geometry, Complex differential geometry and nonlinear differential equation, Amer. Math. Soc., Providence, RI, Contemp. Math. ed., 1986. (1986) MR0833809
  17. Siu, Y.T., 10.2307/1971321, Ann. of Math. (2) 112 (1) (1980), 73–111. (1980) MR0584075DOI10.2307/1971321
  18. Tasayco, D., Zhou, D., 10.1007/s00013-017-1057-9, Arch. Math. (Basel) 109 (2) (2017), 191–200. (2017) MR3673637DOI10.1007/s00013-017-1057-9
  19. Varopoulos, N.Th., Potential theory and diffusion of Riemannian manifolds, Conference on Harmonic Analysis in honor of Antoni Zygmund, vol. I, II, Wadsworth Math. Ser., Wadsworth, Belmont, Calif., 1983, pp. 821–837. (1983) MR0730112
  20. Vieira, M., 10.1007/s00013-013-0594-0, Arch. Math. (Basel) 101 (2013), 581–590. (2013) MR3133732DOI10.1007/s00013-013-0594-0

NotesEmbed ?

top

You must be logged in to post comments.