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Convex functions on complete noncompact manifolds: Topological structure

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References

  1. Bangert, V.: Riemannsche Mannigfaltigkeiten mit nicht-konstanter konvexer Funktionen. Archiv der Math.31, 163–170 (1978)

    Google Scholar 

  2. Bangert, V.: Totally convex sets in complete Riemannian manifolds, Preprint

  3. Bishop, R., O'Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc.145, 1–49 (1969)

    Google Scholar 

  4. Busemann, H.: The geometry of geodesics. New York: Academic Press, 1955

    Google Scholar 

  5. Cheeger, J., Gromoll, D.: On the structure of complete manifods on nonnegative curvature. Ann. of Math.96, 413–443 (1972)

    Google Scholar 

  6. Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Diff. Geometry6, 119–128 (1971)

    Google Scholar 

  7. Greene, R.E., Shiohama, K.: Riemannian manifolds having a nowhere constant convex function. Notices Amer. Math. Soc. Vol.26, No. 2 a-223 (1979)

    Google Scholar 

  8. Greene, R.E., Shiohama, K.: Convex functions on complete noncompact manifolds: Differentiable structure, to appear

  9. Greene, R.E., Wu, H.: On the subharmonicity and plurisubharmonicity of geodesically convex functions. Indiana Univ. Math. J.22, 641–653 (1973)

    Google Scholar 

  10. Greene, R.E., Wu, H.: Integrals of subharmonic functions on manifolds of nonnegative curvature. Inventiones math27, 265–298 (1974)

    Google Scholar 

  11. Greene, R.E., Wu, H.:C convex functions and manifolds of positive curvature. Acta Math.137, 209–245 (1976)

    Google Scholar 

  12. Greene, R.E., Wu, H.:C approximation of convex, subharmonic, and plurisubharmonic functions. Ann. scient. Ec. Norm. Sup. 4e serie t.12, 47–84 (1979)

    Google Scholar 

  13. Gromoll, D., Meyer, W.: On complete open manifolds of positive curvature. Ann. of Math.90, 75–90 (1969)

    Google Scholar 

  14. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, II. New York: Wiley-Interscience, 1969

    Google Scholar 

  15. Narasimhan, R.: Analysis on real and complex manifolds. Amsterdam: North-Holland, 1968

    Google Scholar 

  16. Poor, W.A.: Some results on nonegatively curved manifolds. J. Diff. Geometry9, 583–600 (1974)

    Google Scholar 

  17. Walter, R.: A generalized Allendoerfer-Weil formula and an inequality of the Cohn-Vossen type. J. Diff. Geometry10, 167–180 (1975)

    Google Scholar 

  18. Walter, R.: On the metric projection onto convex sets in Riemannian spaces. Archiv d. Math.25, 91–98 (1974)

    Google Scholar 

  19. Whitehead, J.H.C.: Convex regions in the geometry of paths. Quartery J. Math.2(3), 33–42

  20. Yau, S.T.: Nonexistence of continuous convex functions on certain Riemannian manifolds. Math. Ann.207, 269–270 (1974)

    Google Scholar 

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Research supported by the National Science Foundation, the Institute for Advanced Study and an Alfred P. Sloan Foundation Fellowship (Greene), and by the University of California, Los Angeles (both authors) and by the Sonderforschungsbereich Theoretische Mathematik, Universität Bonn (both authors)

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Greene, R.E., Shiohama, K. Convex functions on complete noncompact manifolds: Topological structure. Invent Math 63, 129–157 (1981). https://doi.org/10.1007/BF01389196

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