The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold

Author(s)
Huashui Zhan

ABSTRACT

Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M, a, b are constants and . Then there exist harmonic functions on M .

Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M, a, b are constants and . Then there exist harmonic functions on M .

Cite this paper

nullH. Zhan, "The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold,"*Advances in Pure Mathematics*, Vol. 1 No. 2, 2011, pp. 5-8. doi: 10.4236/apm.2011.12003.

nullH. Zhan, "The Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold,"

References

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[6] Y. Macrognathia, “Complete Open Manifolds of Nonnegative Radial Curvature,” Pacific Journal of Mathematics, Vol. 165, No. 1, 1994, pp. 153-160.

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[1] S. T. Yau and R. Scheon, “Differential Geometry,” Science Publishing Company of China, Beijing, 1988, p. 37.

[2] M. T. Anderson and R. Scheon, “Positive Harmonic Functions on Complete Manifolds of Negative Curvature,” Annals of Mathematics, Vol. 121, No. 3, 1985, pp. 429- 461. doi:10.2307/1971181

[3] D. Sullivan, “The Ditchless Prob-lem at Infinity for a Negatively Curved Manifold,” Journal of Differential Geometry, Vol. 18, 1983, pp. 723-732.

[4] W. Klingenberg, “Riemannian Geometry,” De Grunter, Berlin, 1982.

[5] Y. Macrognathia, “Manifolds with Pinched Radial Curvature,” Proceedings of the American Mathematical Society, Vol. 118, No. 3, 1993, pp. 975-985.

[6] Y. Macrognathia, “Complete Open Manifolds of Nonnegative Radial Curvature,” Pacific Journal of Mathematics, Vol. 165, No. 1, 1994, pp. 153-160.

[7] C. Y. Xia, “Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth,” Commentarii Mathematici Helvetici, Vol. 74, No. 3, 1999, pp. 456-466. doi:10.1007/s000140050099