Correction to: Comparison Geometry for Integral Bakry–Émery Ricci Tensor Bounds

1 Correction to: J Geom Anal (2019) 29:828–867 https://doi.org/10.1007/s12220-018-0020-8

Based on the recent joint work [1], we correct an error in Theorems 1.1 and 3.1 for the case $\frac{\pi }{2\sqrt{H}}<r<\frac{\pi }{\sqrt{H}}$, $H>0$, and Theorem 1.6 of [2]. The detailed description and further development will appear in [1].

In Theorem 1.1, for the case $\frac{\pi }{2\sqrt{H}}<r<\frac{\pi }{\sqrt{H}}$, $H>0$, we need to add an additional condition:

$$\begin{aligned} \partial _rf=-a\quad \mathrm {or}\quad \partial _rf\ge -a-2(n-1)\sqrt{H}\cot (\sqrt{H}r). \end{aligned}$$

(1)

Because in the proof of this case (see page 837 in [2]), we overlooked the negative property of $m_H$ from

$$\begin{aligned} \varphi '+\frac{1}{n-1}\left[ (\varphi +a+\partial _rf)(\varphi +2m_H+a+\partial _rf)\right] \le {{\mathrm {Ric}^H_f}_-} \end{aligned}$$

to

$$\begin{aligned} \varphi '+\frac{\varphi ^2}{n-1}+\frac{2m_H\varphi }{n-1}\le {{\mathrm {Ric}^H_f}_-}\,. \end{aligned}$$

To correct this error, we add an additional condition (1) to ensure the above argument still holds.

The correction impacts Theorem 3.1 for the case $\frac{\pi }{2\sqrt{H}}<r\le R<\frac{\pi }{\sqrt{H}}$, $H>0$ in [2]. We should also add the additional condition (1) to ensure Theorem 3.1 remains true for this case.

The correction also impacts the assumption on f in Theorem 1.6 of [2]. The correct statement should be

Theorem 1

(Theorem 1.6 [2]). Let $(M,g,e^{-f}dv)$ be an n-dimensional smooth metric measure space. Given $p>n/2$, $a\ge 0$, $H>0$ and $R>0$, there exist $D=D(n,H,a)$ and $\epsilon =\epsilon (n,p,a,H,R)$ such that if $\bar{k}(p,H,a,R)<\epsilon$ and

$$\begin{aligned} \partial _rf=-a\quad \mathrm {or}\quad \partial _rf\ge -a-2(n-1)\sqrt{H}\cot (\pi -\bar{k}(p,H,a,R)) \end{aligned}$$

along all minimal geodesic segments from any $x\in M$, then $diam (M)\le D$.

We remark that the proofs of these results are the same as the previous arguments except using the present condition instead of the previous condition.