A note on the Yamabe problem of Randers metrics

Document Type : Original Article


School of Mathematical Sciences, Tongji University, Shanghai, China, 200092


The classical Yamabe problem in Riemannian geometry states that every conformal class contains a metric with constant scalar curvature. In Finsler geometry, the C-convexity is needed in general. In this paper, we study the strong C-convexity of Randers metrics, and provide a result on the Yamabe problem for the metrics of Randers type.


Main Subjects

[1] T. Aubin, Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296.
[2] D. Bao, S. -S. Chern and Z.Shen, An introduction to Riemann-Finsler geometry, Grad. Texts in Math. 200, Springer, New York, 2000.
[3] B. Chen and Y. Shen, On a class of critical Riemann-Finsler metrics, Publ. Math. Debrecen, 72/3-4 (2008), 451-468.
[4] X. Cheng and M. Yuan, On Randers metrics of isotropic scalar curvature, Publ. Math. Debrecen, 84/1-2 (2014), 63-74.
[5] B. Chen and L. Zhao, On a Yamabe type problem in Finsler geometry, Canad. Math. Bull. 60(2) (2017), 253-268.
[6] B. Chen and L. Zhao, Finsler conformal changes preserving the modified Ricci curvature, Tohoku Math. J., 73 (2021), 39-48.
[7] Q. He and Y. Shen, Some results on harmonic maps for Finsler manifolds, Inter. Jour. of Math., Vol.16, No.9(2005), 1017-1031.
[8] M. Ji and Z. Shen, On strongly convex graphs in Minkowski geometry, Canad. Math. Bull. 45(2) (2002), 232-246.
[9] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479-495.
[10] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa 22 (1968), 265-274.
[11] H. Yamabe, On the deformation of Riemannian structures on compact manfolds, Osaka Math. Jour., 12 (1960), 21-37.