Mean curvature comparison for tubular hypersurfaces in Kähler manifolds and some applications

Vicente Miquel; Vicente Palmer

Compositio Mathematica (1993)

  • Volume: 86, Issue: 3, page 317-335
  • ISSN: 0010-437X

How to cite


Miquel, Vicente, and Palmer, Vicente. "Mean curvature comparison for tubular hypersurfaces in Kähler manifolds and some applications." Compositio Mathematica 86.3 (1993): 317-335. <>.

author = {Miquel, Vicente, Palmer, Vicente},
journal = {Compositio Mathematica},
keywords = {sectional curvatures; Jacobi operator; totally geodesic; unit Hopf bundle; mean curvatures; first Dirichlet eigenvalue; mean exit time},
language = {eng},
number = {3},
pages = {317-335},
publisher = {Kluwer Academic Publishers},
title = {Mean curvature comparison for tubular hypersurfaces in Kähler manifolds and some applications},
url = {},
volume = {86},
year = {1993},

AU - Miquel, Vicente
AU - Palmer, Vicente
TI - Mean curvature comparison for tubular hypersurfaces in Kähler manifolds and some applications
JO - Compositio Mathematica
PY - 1993
PB - Kluwer Academic Publishers
VL - 86
IS - 3
SP - 317
EP - 335
LA - eng
KW - sectional curvatures; Jacobi operator; totally geodesic; unit Hopf bundle; mean curvatures; first Dirichlet eigenvalue; mean exit time
UR -
ER -


  1. [Bo] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York, 1986. Zbl0596.53001MR861409
  2. [BC] R.L. Bishop and R.H. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. Zbl0132.16003MR169148
  3. [Cg] S.Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z.143 (1975), 289-297. Zbl0329.53035MR378001
  4. [Ch] I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984. Zbl0551.53001MR768584
  5. [CV] B.Y. Chen and L. Vanhecke, Differential geometry of geodesic spheres, J. Reine Angew. Math.325 (1981), 28-67. Zbl0503.53013MR618545
  6. [CY] J. Cheeger and S.T. Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math.34 (1981), 465-480. Zbl0481.35003MR615626
  7. [Dy] E.B. Dynkin, Markov processes, Springer Verlag, 1965. Zbl0132.37901
  8. [EH] J.H. Eschenburg and E. Heintze, Comparison theory for Riccati equations, Manuscripta Math.68 (1990), 209-215. Zbl0714.53029MR1063226
  9. [Ga] M. Gage, Upper bounds for the first eigenvalue of the Laplace-Beltrami operator, Indiana Univ. Math. J.29 (1980), 897-912. Zbl0465.53031MR589652
  10. [Gi1] F. Giménez, Comparison theorems for the volume of a complex submanifold of a Kähler manifold, Israel J. Math.71 (1990), 239-255. Zbl0731.53062MR1088818
  11. [Gi2] F. Giménez, Teoremas de comparacin en variedades Kählerianas," Tesis Doctoral. Departamento de Geometria y Topologia. Universidad de Valencia, 1990. 
  12. [Gr1] A. Gray, Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Topology21 (1982), 201-228. Zbl0487.53035MR642000
  13. [Gr2] A. Gray, Volumes of tubes about complex submanifolds of complex projective space, Trans. Amer. Math. Soc.291 (1985), 437-449. Zbl0575.53043MR800247
  14. [Gr3] A. Gray, Tubes, Addison-Wesley, Reading, 1990. Zbl0692.53001MR1044996
  15. [GKP] A. Gray, L. Karp and M.A. Pinsky, The Mean Exit Time from a Tube in a Riemannian Manifold, in (1986), Probability Theory and Harmonic Analysis, edited by J. A. Chao and W. A. Woyczyński, Marcel Dekker, New York. Zbl0593.58047MR830234
  16. [GM1] F. Giménez and V. Miquel, Volume estimates for real hypersurfaces of a Kähler manifold with strictly positive holomorphic sectional and antiholomorphic Ricci curvatures, Pacific J. Math.142 (1990), 23-39. Zbl0728.53034MR1038727
  17. [GM2] F. Giménez and V. Miquel, Bounds for the first Dirichlet eigenvalue of domains in Kähler manifolds, Archiv der Math.56 (1991), 370-375. Zbl0735.58009MR1094424
  18. [GW] R.E. Greene and H. Wu, C∞ approximations of convex, subharmonic and plurisubharmonic functions, Ann. Sci. Ec. Nor. Sup.12 (1979), 47-84. Zbl0415.31001
  19. [HK] E. Heintze and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Ec. Nor. Sup.11 (1978), 451-470. Zbl0416.53027MR533065
  20. [Ka] H. Karcher, Riemannian comparison constructions, in, Global Differential Geometry, edited by S. S. Chern, M.A.A., 1989, 170-222. Zbl0683.53040MR1013810
  21. [Ki] M. Kimura, Real Hypersurfaces and complex manifolds in complex projective space, Trans. Am. Math. Soc.296 (1986), 137-149. Zbl0597.53021MR837803
  22. [Ks] A. Kasue, On a lower bound for the first eigenvalue of the Laplace operator on a Riemannian manifold, Ann. Sci. Ec. Norm. Sup.17 (1984), 31-44. Zbl0553.53026MR744066
  23. [K-N] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience, New York, 1969. Zbl0175.48504
  24. [Le] J.M. Lee, Eigenvalue comparison for tubular domains, Proc. Am. Math. Soc.109 (1990), 843-848. Zbl0702.58076MR1021900
  25. [Ma] S.M. Markvorsen, On the heat kernel comparison theorems for minimal submanifolds, Proc. Amer. Math. Soc.97 (1986), 479-482. Zbl0597.53049MR840633
  26. [MP] V. Miquel and V. Palmer, A comparison theorem for the mean exit time from a domain in a Kähler manifold, Ann. Global Anal. Geom, 10 (1992) 73-80. Zbl0773.53027MR1172621
  27. [Na] S. Nayatani, On the volume of positively curved Kähler manifolds, Osaka J. Math.25 (1988), 223-231. Zbl0647.53050MR937197
  28. [Pa] V. Palmer, Un teorema de comparación para el tiempo de salida medio de una bola geodésica en una variedad Kähler, XV Jornadas Luso-espanholas de Matemática (1990) III, 173-178. MR1161810
  29. [Ro] H.L. Royden, Comparison Theorems for the Matrix Riccati Equation, Comm. on Pure and Applied Math.41 (1988), 739-746. Zbl0632.34028MR948079
  30. [Sz] Z.I. Szabó, The Lichnerowicz conjecture on Harmonic manifolds, J. Differential Geom.31 (1990), 1-28. Zbl0686.53042MR1030663

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.