Abstract
Given a complete Riemannian manifold (M, g) with nonnegative sectional curvature outside a compact subset. Let h be another Riemannian metric which is uniformly equivalent to g. It was shown that the dimension of the space of bounded harmonic functions on (M, h) is finite and is the same as of that under metric g, and the dimension of the space spanned by nonnegative harmonic functions on (M, h) is also finite and is the same as of that under metric g. Moreover, bases were constructed for both spaces on (M, h) and precise estimates were established on the asymptotic behavior at infinity for those basic functions.
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References
Abresch, U. Lower curvature bounds, Toponogov’s theorem, and bounded topology,Ann. Sci. Ecole Norm. Sup.,18, 651–670, (1985).
Axler, A., Bourdon, P., and Ramey, W. Harmonic Function Theory, Springer-Verlag, New York, 1992.
Cheng, S.Y. and Yau, S.T. Differential equations on Riemannian manifolds and their geometric applications,Comm. Pure Appl. Math.,28, 333–354, (1975).
Donnelly, H. Bounded harmonic functions and positive Ricci curvature,Math. Z,191, 559–565, (1986).
Grigor’yan, A.A. A criterion for the existence on a Riemannian manifold of a nontrivial bounded harmonic function with finite Dirichlet integral,Soviet Math. Dokl.,35, 339–341, (1987).
Grigor’yan, A.A. On the existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds,Math. USSR-Sbornik,56, 349–358, (1987).
Gilbarg, D. and Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 2nd edition, 1983.
Kasue, A. A compactification of a manifold with asymptotically nonnegative curvature,Ann. Sci. Ecole. Norm. Sup.,21, 593–622, (1988).
Kasue, A. Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I. Preprint.
Li, P. On the structure of complete Kahler manifolds with nonnegative curvature near infinity,Invent. Math.,99, 576–600, (1990).
Li, P. Lecture Notes on the Geometric Analysis, Research Inst. of Math. Global Analysis Research Center, Seoul National Univ., Lecture notes series 6, (1990).
Liu, Z-d. Ball covering on manifolds with nonegative Ricci curvature near infinity. Preprint.
Lyons, T.J. Instability of the Liovuille property for quasi-isometric Riemannian manifolds and reversible Markov chains,J. Differential Geom.,26, 33–66, (1987).
Lettman, W., Stampcchia, G., and Weinberger, H.F. Regular points for elliptic equations with discontinuous coefficients,Ann. Scuola Norm. Sup. Pisa,17(3), 43–77, (1960).
Li, P. and Tarn, L. Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set,Annals of Math.,125, 171–207, (1987).
Li, P. and Tam, L. Harmonic functions and the structure of complete manifolds,J. Differential Geom.,35, 359–383, (1992).
Li, P. and Tam, L. Symmetric Green’s functions on complete manifolds,Amer. J. Math.,109, 1129–1154, (1987).
Li, P. and Yau, S.T. On the parabolic kernel of the Schrodinger operator,Acta Math.,156, 153–301, (1986).
Royden, H. The growth of a fundamental solution of an elliptic divergence structure equation,Studies in Math. Analysis and Related topics: Essays in Honor of George Polya, Stanford, 333–340, (1962).
Saloff-Coste, L. Uniformly elliptic operators on Riemannian manifolds,J. Differential Geom., (1992).
Tam, L. Harmonic functions on connected sums of manifolds,Math. Z,211, 315–322, (1992).
Tam, L. Lecture notes on harmonic functions, 1993.
Yau, S.T. Harmonic functions on complete Riemannian manifolds,Comm. Pure Appl. Math.,28, 201–228, (1975).
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Sung, CJ. Harmonic functions under quasi-isometry. J Geom Anal 8, 143–161 (1998). https://doi.org/10.1007/BF02922112
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DOI: https://doi.org/10.1007/BF02922112