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Gradient Estimates for Harmonic Functions on Manifolds With Lipschitz Metrics

Published online by Cambridge University Press:  20 November 2018

Jingyi Chen  and
Elton P. Hsu
Jingyi Chen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A.
Elton P. Hsu
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois 60208, U.S.A. email: [email protected]
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Abstract

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We introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

Keywords

60D5828D05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 6 , 01 December 1998 , pp. 1163 - 1175
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Anderson, M. and Cheeger, J., Cα-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differential Geom. 35(1992), 265–281.Google Scholar
2. Cheng, S.Y., Eigenvalue comparison theorems and its geometric applications. Math. Z. 143(1975), 289–297.Google Scholar
3. Colding, T.H., Ricci curvature and volume convergence. (1995), preprint.Google Scholar
4. Colding, T.H., Large manifolds with positive Ricci curvature. (1995), preprint.Google Scholar
5. Donnelly, H., Bounded harmonic functions and positive Ricci curvature. Math. Z. 191(1986), 559–565.Google Scholar
6. Gilbarg, D. and Trudinger, N., Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1983.Google Scholar
7. Green, R. and Wu, H., Lipschitz convergence of Riemannian manifolds. Pacific J. Math. 131(1988), 119–141.Google Scholar
8. Hörmander, L., The Analysis of Linear Partial Differential Operators I. Springer-Verlag, Berlin, 1983.Google Scholar
9. Lin, F.H., Asymptotic conic elliptic operators and Liouville type theorems. (1995), preprint.Google Scholar
10. Li, P. and Tam, L.F., Harmonic functions and the structure of complete manifolds. J. Differential Geom. 35(1992), 359–383.Google Scholar
11. Li, P., Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set. Ann. of Math. (2) 125(1987), 171–207.Google Scholar
12. Yau, S.-T., Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28(1975), 201–228.Google Scholar
13. Yau, S.-T., Survey on partial differential equations in differential geometry. Ann. of Math. Study 102(1982), 3–73.Google Scholar