Abstract
In the last three chapters of Part II, having the fundamental properties of nonlinear and linearized heat semigroups in the previous chapter in hand, we present three kinds of applications of the Bochner inequality by generalizing the Γ-calculus to the Finsler setting (the so-called nonlinear Γ-calculus).
Throughout these three chapters, we assume the compactness of M to avoid delicate technical issues. In this chapter, we show the L 2- and L 1-gradient estimates followed by the Li–Yau gradient estimate and Harnack inequality. We will also see that the L 2- and L 1-gradient estimates are both equivalent to the lower weighted Ricci curvature bound \( \mathop {\mathrm {Ric}} \nolimits _{\infty } \ge K\).
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Ohta, Si. (2021). Gradient Estimates. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_14
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DOI: https://doi.org/10.1007/978-3-030-80650-7_14
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