In this paper, we give an (integrated) \(p(>1)\)-Bochner–Weitzenböck formula and the p-Reilly type formula on Finsler manifolds. As applications, we obtain the p-Poincaré inequality on an n-dimensional compact Finsler manifold without boundary or with convex boundary under the assumption that Ricci\(_N\ge K\) for \(N\in [n, \infty ]\) and \(K\in \mathbb {R}\). In fact, we have the sharp p-Poincaré inequality on such a manifold. Based on this, we further prove the existence of two types of optimal \((\mathfrak p, \mathfrak q)\)-Sobolev inequalities on compact Finsler manifolds with Ric\(_N\ge K\) for \(N\in [n, \infty )\) and \(K\in \mathbb {R}\). In particular, when \(K>0\), we establish the sharp \((\mathfrak p, 2)\)-Sobolev inequality for \(2\le \mathfrak p\le 2N/(N-2)\). With this, we obtain the sharp lower bound estimate for the first eigenvalue \(\lambda _1\) of the Finsler Laplacian and characterize the manifold on which \(\lambda _1\) reaches its extremum, generalizing the well known Lichnerowicz–Obata’s results (in: Geometrie des groupes de transformations, Dunod, Paris, 1958; J Math Soc Jpn 14:333–340, 1962).

中文翻译：

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Finsler流形上的几何和函数不等式

在本文中，我们在Finsler流形上给出了一个（积分）\（p（> 1）\）- Bochner–Weitzenböck公式和p -Reilly型公式。作为应用，在Ricci \（_ N \ ge K \）为\（N \ in [n，\ infty] \的情况下，我们在无边界或具有凸边界的n维紧Finsler流形上获得p- Poincaré不等式）和\（K \ in \ mathbb {R} \）中。实际上，在这样的流形上，我们有很明显的p- Poincaré不等式。在此基础上，我们进一步证明了带有Ric的紧型Finsler流形上两种最优\（（（mathfrak p，\ mathfrak q）\）- Sobolev不等式的存在\（_ N \ ge K \）表示\（N [in，[infty）\）和\（K \ in \ mathbb {R} \）。特别地，当\（K> 0 \）时，我们为\（2 \ le \ mathfrak p \ le 2N /（N-2）\）建立了尖锐的\（（（mathfrak p，2）\）- Sobolev不等式。这样，我们就获得了Finsler Laplacian的第一个特征值\（\ lambda _1 \）的下界估计值，并刻画了\（\ lambda _1 \）达到极值的流形，从而概括了众所周知的Lichnerowicz–Obata的结果（在：Deood，巴黎，1958年；《几何的变形》，1962年，J Math Soc Jpn 14：333-340）。