Bull. Korean Math. Soc. 2024; 61(2): 433-450
Online first article March 13, 2024 Printed March 31, 2024
https://doi.org/10.4134/BKMS.b230136
Copyright © The Korean Mathematical Society.
Huiting Chang, Fanqi Zeng
Xinyang Normal University; Xinyang Normal University
Using a Reilly type integral formula due to Li and Xia \cite{LiXia2017}, we prove several geometric inequalities for affine connections on Riemannian manifolds. We obtain some general De Lellis-Topping type inequalities associated with affine connections. These not only permit to derive quickly many well-known De Lellis-Topping type inequalities, but also supply a new De Lellis-Topping type inequality when the $1$-Bakry-\'{E}mery Ricci curvature is bounded from below by a negative function. On the other hand, we also achieve some Lichnerowicz type estimate for the first (nonzero) eigenvalue of the affine Laplacian with the Robin boundary condition on Riemannian manifolds.
Keywords: Reilly type formula, affine connection, De Lellis-Topping inequality, eigenvalue
MSC numbers: Primary 53C21; Secondary 58J32
Supported by: The research of authors is supported by NSFC (No.12101530), the Science and Technology Project of Henan Province (No.232102310321), and the Key Scientific Research Program in Universities of Henan Province (Nos.21A110021, 22A110021) and Nanhu Scholars Program for Young Scholars of XYNU (No.2019).
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