Titlebook: Comparison Finsler Geometry; Shin-ichi Ohta Book 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer

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The Bochner–Weitzenb?ck Formulae Bochner inequality can be regarded as a nonlinear analogue of the Γ.-criterion..Coupled with the nonlinear heat flow discussed in the next chapter, the Bochner inequality has fruitful applications including gradient estimates (Chap. .), isoperimetric inequalities (Chap. .), and functional inequalities (Chap. .).

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Gradient Estimatest these three chapters, we assume the compactness of . to avoid delicate technical issues. In this chapter, we show the ..- and ..-. followed by the . and .. We will also see that the ..- and ..-gradient estimates are both equivalent to the lower weighted Ricci curvature bound ..

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Curvatureads us to a useful and inspiring description of the Finsler curvature as the Riemannian curvature of ... The metric .. is also called an ., and its application to the Riemannian characterization of the Finsler curvature goes back to Ottó Varga.

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