Laplacian on complex Finsler manifolds

Abstract

In this paper, the Laplacian on the holomorphic tangent bundle T ^1,0 M of a complex manifold M endowed with a strongly pseudoconvex complex Finsler metric is defined and its explicit expression is obtained by using the Chern Finsler connection associated with (M, F). Utilizing the initiated “Bochner technique”, a vanishing theorem for vector fields on the holomorphic tangent bundle T ^1,0 M is obtained.

Keywords

Laplacian / Strongly pseudoconvex complex Finsler metric / Chern Finsler connection

References

[1] Bochner S.. Vector fields and Ricci curvature. Bull. Amer. Math. Soc., 1946, 52: 776-797

[2] Bochner S.. Curvature in Hermitian metric. Bull. Amer. Math. Soc., 1947, 53: 179-195

[3] Bochner S.. Curvature and Betti numbers I, II. Ann. of Math., 1948, 49: 379-390

[4] Yano K., Bochner S.. Curvature and Betti Numbers, 1953, New Jersey: Princeton University Press

[5] Wu H.-H.. The Bochner Technique in Differential Geometry, 1988, London: Harwood Academic Publishers

[6] Morrow J., Kodaira K.. Complex Manifold, Holt, 1971, New York: Rinehart and Winston, Inc.

[7] Antonelli P. L., Lackey B.. The Theory of Finslerian Laplacians and Applications, MAIA, 459, 1998, Dordrecht: Kluwer Academic Publishers

[8] Bao D., Lackey B.. Randers surfaces whose Laplacian have completely positive symbol. Nonlinear Analysis, 1999, 38: 27-40

[9] Bao D., Lackey B.. A Hodge decomposition theorem for Finsler spaces. C. R. Math. Acad. Sci. Paris, 1996, 323(1): 51-56

[10] Chern S. S., Shen Z.. An Introduction to Riemannian-Finsler Geometry, 2000, New York: Springer-Verlag

[11] Munteanu O.. Weitzenböck formulas for horizontal and vertical Laplacians. Houston J. Math., 2003, 29(4): 889-900

[12] Yan R.. Laplace operator on Finsler manifold. Acta Math. Sci., 2004, 24A(4): 420-425

[13] Zhong C. P., Zhong T. D.. Horizontal \bar \partial-Laplacian on complex Finsler manifolds. Sci. China Ser. A, 2005, 48(Supp.): 377-391

[14] Zhong C. P., Zhong T. D.. Hodge decomposition theorem on strongly Kähler Finsler manifolds. Sci. China Ser. A, 2006, 49(11): 1696-1714

[15] Abate M., Patrizio G.. Finsler Metric—A Global Approach. Lect. Notes in Math., 1591, 1994, Berlin, Heidelberg: Springer-Verlag

[16] Zhong C. P.. Laplacians on the holomorphic tangent bundle of a Kaehler manifold. Sci. China Ser. A, 2009, 52(12): 2841-2854

[17] Zhong C. P.. A vanishing theorem on Kaehler Finsler manifolds. Diff. Geom. Appl., 2009, 27: 551-565

[18] Kodaira K.. On a differential-geometric method in the theory of analytic stacks. Proc. Natl. Acad. Sci. USA, 1953, 39: 1268-1273

[19] Siu Y.-T.. The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds. Ann. of Math., 1980, 112: 73-111

[20] Siu Y.-T.. Complex analyticity of harmonic maps, vanishing and Lefschetz theorems. J. Diff. Geom., 1982, 17: 55-138

[21] Xiao J. X., Zhong T. D., Qiu C. H.. Bochner technique in strongly Kähler Finsler manifold. Acta Math. Sci., 2010, 30B(1): 89-106

[22] Xiao, J. X., Zhong, T. D. and Qiu, C. H., Bochner-Kodaira techniques on Kähler Finsler manifolds, J. Geom. Anal., submitted.

[23] Munteanu O.. Complex Spaces in Finsler, Lagrange and Hamilton Geometries, 2004, Dordrecht: Kluwer Academic Publishers

[24] Pitis G., Munteanu O.. v-Cohomology of complex Finsler manifolds. Stud. Univ. Babes-Bolyai Math., 1998, 18(3): 889-900