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On a Yamabe Type Problem in Finsler Geometry

  • Bin Chen (a1) and Lili Zhao (a2)

Abstract

In this paper, a newnotion of scalar curvature for a Finsler metric $F$ is introduced, and two conformal invariants $Y(M,F)$ and $C(M,F)$ are defined. We prove that there exists a Finsler metric with constant scalar curvature in the conformal class of $F$ if the Cartan torsion of $F$ is sufficiently small and $Y(M,F)C(M,F)<Y({{\mathbb{S}}^{n}})$ where $Y({{\mathbb{S}}^{n}})$ is the Yamabe constant of the standard sphere.

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[1] Akbar-Zadeh, H., Generalized Einstein manifolds J. Geom. Phys. 17(1995), 342380. http://dx.doi.org/1 0.101 6/0393-0440(94)00052-2
[2] Aubin, T., Problèmes isoperimetriques et espaces de Sobolev. J. Diff. Geom. 11(1976), 533598.
[3] Aubin, T., Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl. 55(1976), 269296.
[4] Bao, D., On two curvature-driven problems in Riemann-Finsler geometry. Finsler Geometry, Sapporo 2005-In Memory of Makoto Matsumoto, Advanced Studies in Pure Mathematics: Volume 48.
[5] Bao, D., Chern, S.-S., and Shen, Z., An introduction to Riemann-Finsler geometry. Grad. Texts in Math. 200, Springer, New York, 2000.
[6] Basco, S. and Cheng, X., Finsler conformai transformations and the curvature invariance. Publ. Math. Debrecen, 70/1-2(2007), 221231.
[7] Chen, B. and Shen, Y., On a class of critical Riemann-Finsler metrics, Publ. Math. Debrecen, 72/3-4(2008), 451468.
[8] Cheng, X. and Yuan, M., On Randers metrics ofisotropic scalar curvature. Publ. Math. Debrecen, 84/1-2(2014), 6374. http://dx.doi.org/10.5486/PMD.2O1 4.5833
[9] He, Q. and Shen, Y., Some results on harmonic maps for Finsler manifolds. Inter. J. Math, 16(2005), no. 9, 10171031. http://dx.doi.org/10.1142/S012 9167X05003211
[10] Lee, J. and Parker, T., The Yamabe problem, Bull. Amer. Math. Soc. 17(1987), 3791. http://dx.doi.org/10.1090/S0273-0979-1987-15514-5
[11] Schoen, R., Conformai deformation of a Riemannian metric to constant scalar curvature. J. Diff. Geom. 20(1984), 479495.
[12] Schoen, R. and Yau, S. T., Lectures on differential geometry. International Press, Boston, 1994.
[13] Trudinger, N., Remarks concerning the conformai deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa 22(1968), 265274.
[14] Yamabe, H., On the deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12(1960), 2137.
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On a Yamabe Type Problem in Finsler Geometry

  • Bin Chen (a1) and Lili Zhao (a2)

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