Corrigendum to“Nonlinear geometric analysis on Finsler manifolds”[Eur. J. Math. 3 (2017), 916–952]
Shin-ichi Ohta∗

Abstract
We list corrigenda to the survey article [4], due to the revisions of the articles[2, 3] whose results were reviewed in [4].

First, related to [2], we have the following corrigenda concerning the noncompact case.
• In the integrated form of the Bochner inequality [4, Theorem 3.5], we need to employ
21u ∈ Hloc
(M ) ∩ C 1 (M ) such that ∆u ∈ Hloc
(M ) and ϕ ∈ Hc1 (M ) ∩ L∞ (M ) as in
the original paper [6, Theorem 3.6] (see also [2, Theorem 2.13], [5, Theorem 12.13]).
We consider u and ϕ in the same classes also in the improved Bochner inequality[4, Proposition 4.4] (see [2, Corollary 3.6], [5, Corollary 12.16]).
• Among the properties of linearized heat semigroups in [4, Proposition 4.1(ii)], theHölder continuity holds true and is enough for our applications. We refer to [2,Proposition 3.1] or [5, Proposition 13.20] for details.
• Due to the restriction of the class of admissible functions in the integrated Bochner
inequality as above, we need some additional assumptions in the gradient estimates[4, Theorems 4.3, 4.5]. Precisely, we assume that (M, F, m) satisfies Ric∞ ≥ K,CF < ∞ and SF < ∞, and that (ut )t≥0 is a global solution to the heat equation
satisfying u0 ∈ Cc∞ (M ) and()d[F (∇ut )] ∇∇ut [F (∇ut )] ∈ L1 (M )(1)for all t > 0 (see [2, Theorem 3.7, Corollary 3.8]). We refer to [2, §3.4] for a furtherdiscussion on the (likely redundant) assumption (1).
• Similarly, in the characterizations of Ric∞ ≥ K in [4, Theorem 4.6] as well as theisoperimetric inequality [4, Theorem 6.3], we need to assume CF < ∞, SF < ∞,and (1) for all solutions (ut )t≥0 to the heat equation with u0 ∈ Cc∞ (M ) (see [2,Theorems 3.9, 4.1]).

∗Department of Mathematics, Osaka University, Osaka 560-0043, Japan (s.ohta@math.sci.osaka-u.ac.jp)

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Next, in the Sobolev inequality [4, Theorem 5.11] from [3], as explained in the erratumof [3], the case of 1 ≤ p < 2 should be discussed separately and called the Beckner
inequality. The statement itself of [4, Theorem 5.11] is true and, moreover, one can also
show the following generalization to N < 0 along the lines of [1] (see [5, Theorem 16.8]
for details).
Theorem 1 Assume that (M, F, m) is compact and satisfies RicN ≥ K > 0 for someN ∈ (−∞, −2) and m(M ) = 1. Then we have∫∥f ∥2Lp − ∥f ∥2L2N −1F 2 (∇f ) dm
≤p−2
KN Mfor all 1 ≤ p ≤ (2N 2 + 1)/(N − 1)2 and f ∈ H 1 (M ).

References[1] I. Gentil and S. Zugmeyer, A family of Beckner inequalities under various curvaturedimension conditions. Bernoulli 27 (2021), 751–771.
[2] S. Ohta, A semigroup approach to Finsler geometry: Bakry–Ledoux’s isoperimetricinequality. Comm. Anal. Geom. (to appear). Available at arXiv:1602.00390[3] S. Ohta, Some functional inequalities on non-reversible Finsler manifolds. Proc. IndianAcad. Sci. Math. Sci. 127 (2017), 833–855; Erratum in ibid. 131 (2021), Paper No. 23,2pp.
[4] S. Ohta, Nonlinear geometric analysis on Finsler manifolds. Eur. J. Math. 3 (2017),916–952.
[5] S. Ohta, Comparison Finsler geometry. Springer Monographs in Mathematics.
Springer, Cham, 2021.
[6] S. Ohta and K.-T. Sturm, Bochner–Weitzenböck formula and Li–Yau estimates onFinsler manifolds. Adv. Math. 252 (2014), 429–448.

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