The lower bounds for the first eigenvalue of the biharmonic and \(p\)-biharmonic operators on Finsler manifolds
The main aim of this paper is to obtain lower bounds for the first eigenvalue of the biharmonic and \(p\)-biharmonic operators on Finsler manifold assuming lower bounds for the weighted Ricci curvature of the underlying space. These problems go back to \textit{A. Lichnérowicz} [Géométrie des groupes de transformations. Paris: Dunod (1958; Zbl 0096.16001)], \textit{M. Obata} [J. Math. Soc. Japan 14, 333--340 (1962; Zbl 0115.39302)] and the results of \textit{H. P. McKean} [J. Differ. Geom. 4, 359--366 (1970; Zbl 0197.18003)] about the first eigenvalue of the Laplace-Beltrami operator on Riemannian manifolds. Note that biharmonic eigenvalue problems define clamped plate and buckling problems, whereas \(p\)-biharmonic eigenvalue problems are their generalization to a higher order.\N\NLet \((M,F)\) be an \(n\)-dimensional Finsler manifold and \(X\in \Gamma(TM)\) be any vector field such that \(\|X\|=\sup_MF(X)<\infty\). The proofs of the results of this paper make use of these vector fields with appropriate divergence in the sense that \(\inf_M \operatorname{div}(X)>0\), and the weighted Bochner formula in the Finslerian setting. The Hessian comparison theorem of \textit{S. Yin} et al. [Publ. Math. Debr. 83, No. 3, 385--405 (2013; Zbl 1340.53112)] is also used to give applications of the main theorems.\N\NSimilar results in different contexts have appeared in the literature, see for instance [\textit{A. Abolarinwa} and \textit{A. Taheri}, Complex Var. Elliptic Equ. 67, No. 6, 1379--1392 (2022; Zbl 1518.58011); \textit{A. Abolarinwa} et al., Adv. Difference Equ. 2021, Paper No. 273, 15 p. (2021; Zbl 1494.58012); \textit{S. Pan} and \textit{L. Zhang}, Differ. Geom. Appl. 47, 190--201 (2016; Zbl 1338.35313); \textit{L. Zhang} and \textit{Y. Zhao}, J. Inequal. Appl. 2016, Paper No. 5, 9 p. (2016; Zbl 1330.35275)] and the references cited therein.
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