Comparison Finsler Geometry

This monograph discusses recent developments in comparison geometry and geometric analysis on Finsler manifolds. It derives fundamental geometric and analytic inequalities in the Finsler context,providing an accessible entry point to Finsler geometry for beginners. The book covers topics such as weighted Ricci curvature,nonlinear Laplacian,and heat flow,and includes advanced topics such as the Cheeger-Gromoll splitting theorem and the curvature-dimension condition.

Format: Paperback / softback
Length: 316 pages
Publication date: 10 October 2022
Publisher: Springer Nature Switzerland AG

This monograph delves into the realm of comparison geometry and geometric analysis on Finsler manifolds, presenting recent advancements in this field. By extending the weighted Ricci curvature to the Finsler setting, the author systematically derives fundamental geometric and analytic inequalities, providing an accessible entry point into Finsler geometry for readers new to the area.

The book is organized into three parts. The first part establishes the fundamentals of Finsler geometry, covering topics such as Jacobi fields, curvature tensors, arc length variation formulas, and classical comparison theorems. The second part introduces the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds, enabling the derivation of various formulas and inequalities. These include the Bochner–Weitzenböck formula, gradient estimates, Bakry–Ledoux's Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. The third part explores advanced topics, including a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Geometric descriptions accompany the results, while exercises encourage active engagement.

Comparison Finsler Geometry serves as an invaluable resource for graduate students and researchers in mathematics, particularly those interested in differential geometry, geometric analysis, and Finsler manifolds. While knowledge of differentiable manifold theory and functional analysis is assumed, readers with a background in Riemannian geometry will find their insights readily transferable. This monograph offers a comprehensive and up-to-date exploration of the latest developments in comparison geometry and geometric analysis on Finsler manifolds, making it a valuable addition to the literature.

Weight: 522g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030806521
Edition number: 1st ed. 2021