A class of Finsler measure spaces of constant weighted Ricci curvature

The weight Ricci curvature plays an important role in studying global Finsler geometry. In this paper, we study a class of Finsler measure spaces of constant weighted Ricci curvature. We explicitly construct new families of such complete Finsler measure spaces. In particular, we find an eigenfunction and its eigenvalue for such spaces, generalizing a result previously only known in the case of Gaussian shrinking soliton. Finally, we give necessary and sufficient conditions on the coordinate functions for these spaces to be Euclidean measure spaces.

PDF

___

[1] Bao DW, Chern SS, Shen ZM. An Introduction to Riemann-Finsler Geometry. Springer, 2000.
[2] Cao HD. Recent progress on Ricci soitons. Advanced Lectures in Mathematics 2010; 11: 1-38.
[3] Cao HD, Zhou DT. On complete gradient shrinking Ricci solitons. Journal of Differential Geometry 2010; 85: 175-185.
[4] Cheng X, Zhou DT. Eigenvalue of the drifted Laplacian on complete metric measure spaces. Communications in Contemporary Mathematics 2017; 19 (1): 1650001.
[5] Cianchi A, Musil V, Pick L. Moser inequalities in Gauss space. Mathematische Annalen 2020; 377: 1265-1312.
[6] Eskenazis A, Moschidis G. The dimensional Brunn-Minkowski inequality in Gauss space. Journal of Functional Analysis 2021; 280 (6): 108914.
[7] Ge YX, Shen ZM. Eigenvalues and eigenfunctions of metric measure manifolds. Proceedings of London Mathematical Society 2001; 82: 725-746.
[8] Huang Y. Minkowski problem in Gaussain probability space. International Conference on Geometric Analysis and PDEs, at the Sichuan University on May 1, 2021.
[9] Latala R. On some inequality for Gaussian measures. Proceedings of the Interntional Congress of Mathematicians. Beijing: Higher Education Press, 2002, pp. 813-822.
[10] Ohta S. Finsler interpolation inequality. Calculus of Variations and Partial Differential Equations 2009; 36: 211-249.
[11] Ohta S. Nonlinear geometric analysis on Finsler manifolds. European Journal of Mathematics 2017; 4: 1-37.
[12] Ohta S, Strum K. Bochner-Weitzenbo¨ck formula and Li-Yau estimates on Finsler manifolds. Advances in Mathematics 2014; 252: 429-448.
[13] Shen ZM. Volume comparision and its application in Riemann-Finsler geometry. Advances in Mathematics 1997; 128: 306-328.
[14] Shen ZM. Lectures on Finsler Geometry. World Scientific Publishing Co, Singapore, 2001.
[15] Shen ZM. Conjugate radius and positive scalar curvature. Mathematische Zietschrift 2001; 238, 431-439.
[16] Tabatabaeifar T, Najafi B, Tayebi A. Weighted projective Ricci curvature in Finsler geometry, Mathematica Slovaca, 71(1), 183-198.
[17] Yin ST. Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below. Frontiers of Mathematics in China 2018; 13: 435-448.
[18] Yin ST, Mo XH. Some results on complete Finsler measure spaces. Journal of Mathematical Analysis and Applications 2021; 497: 124846.