Bounds on Harmonic Radius and Limits of Manifolds with Bounded Bakry–Émery Ricci Curvature

Abstract

Under the usual condition that the volume of a geodesic ball is close to the Euclidean one or the injectivity radii is bounded from below, we prove a lower bound of the $C^{\alpha } \cap W^{1, q}$ harmonic radius for manifolds with bounded Bakry–Émery Ricci curvature when the gradient of the potential is bounded. Under these conditions, the regularity that can be imposed on the metrics under harmonic coordinates is only $C^\alpha \cap W^{1,q}$, where $q>2n$ and n is the dimension of the manifolds. This is almost 1-order lower than that in the classical $C^{1,\alpha } \cap W^{2, p}$ harmonic coordinates under bounded Ricci curvature condition (Anderson in Invent Math 102:429–445, 1990). The loss of regularity induces some difference in the method of proof, which can also be used to address the detail of $W^{2, p}$ convergence in the classical case. Based on this lower bound and the techniques in Cheeger and Naber (Ann Math 182:1093–1165, 2015) and Wang and Zhu (Crelle’s J, http://arxiv.org/abs/1304.4490), we extend Cheeger–Naber’s Codimension 4 Theorem in Cheeger and Naber (2015) to the case where the manifolds have bounded Bakry–Émery Ricci curvature when the gradient of the potential is bounded. This result covers Ricci solitons when the gradient of the potential is bounded. During the proof, we will use a Green’s function argument and adopt a linear algebra argument in Bamler (J Funct Anal 272(6):2504–2627, 2017). A new ingredient is to show that the diagonal entries of the matrices in the Transformation Theorem are bounded away from 0. Together these seem to simplify the proof of the Codimension 4 Theorem, even in the case where Ricci curvature is bounded.

Appendix A

In this section, we prove Claim 4 and finish the proof of Theorem 3.6 part a).

Proof of Claim 4

We first show that

(A.1)

Here, as usual, define

$$\begin{aligned} \langle w_j^{k-1}, d v_j^k\rangle =\sum _{a=1}^{k-1}\langle d v_j^a, dv_j^k\rangle dv_j^1\wedge \cdots \wedge \widehat{dv_j^a}\wedge \cdots \wedge dv_j^{k-1}. \end{aligned}$$

Thus,

$$\begin{aligned}&\left| \langle w_j^{k-1}, dv_j^k\rangle \right| ^2\\&\quad =\sum _{a,b=1}^{k-1}\langle d v_j^a, dv_j^k\rangle \langle d v_j^b, dv_j^k\rangle \\&\qquad \times \,\langle dv_j^1\wedge \cdots \wedge \widehat{dv_j^a}\wedge \cdots \wedge dv_j^{k-1}, dv_j^1\wedge \cdots \wedge \widehat{dv_j^b}\wedge \cdots \wedge dv_j^{k-1}\rangle . \end{aligned}$$

Since $(v_j^1, \ldots , v_j^{k-1})$ is an $\epsilon _j(R)$-splitting on $B_R(x_j)$ by Claim 2, it is not hard to see that

(A.2)

Therefore, (A.1) follows from (3.18) and (A.2) immediately.

Notice that $w_j^k=w_j^{k-1}\wedge dv_j^k$, and

$$\begin{aligned} |w_j^k|^2=|w_j^{k-1}|^2|dv_j^k|^2-|\langle w_j^{k-1},dv_j^k\rangle |^2. \end{aligned}$$

From (3.17), we have

This, together with (A.1), implies that

(A.3)

Combining this with the fact that $(v_j^1,\ldots , v_j^{k-1})$ is an $L_j$-drifted $\epsilon _j(R)$-splitting map on $B_R(x_j)$, we have

Therefore, it finishes the proof of Claim 4. $\square$

Recall that

and

$$\begin{aligned} \tilde{v}_j^l=v_j^l,\quad \tilde{v}_j^k=v_j^k+\sum _{l=1}^{k-1}a_j^lv_j^l. \end{aligned}$$

Since $(v_j^1,\ldots ,v_j^k): B_2(x_j)\rightarrow \mathbb {R}^k$ is $L_j$-drifted $C\epsilon$-splitting, and $(v_j^1,\ldots ,v_j^{k-1}): B_2(x_j)\rightarrow \mathbb {R}^{k-1}$ is $L_j$-drifted $\epsilon _j$-splitting, it follows that

Hence,

$$\begin{aligned} |a_j^l|\le C\epsilon . \end{aligned}$$

(A.4)

The above facts and (3.18) imply that

(A.5)

and

(A.6)

Setting

one has

where we have used (3.31), (A.4), and the fact that $(v_j^1, \ldots , v_j^{k-1})$ is an $L_j$-drifted $\epsilon _j$-splitting map. Therefore, by using the similar technique as in Lemma 3.34 in [10] (or mean value inequality), one can get

$$\begin{aligned} \sup _{B_1(x_j)}|d \hat{v}_j^k|\le 1+\epsilon _j,\ \forall 1\le a\le k, \end{aligned}$$

(A.7)

and

(A.8)

Finally, (A.5) and (A.6) give

(A.9)

It is obvious that (A.7), (A.8), and (A.9) together with the fact that $(v_j^1,\ldots ,v_j^{k-1})$ is $L_j$-drifted $\epsilon _j$-splitting imply that $(\hat{v}_j^1,\ldots ,\hat{v}_j^k): B_1(x_j)\rightarrow \mathbb {R}^k$ is an L-drifted $\epsilon _j$-splitting map. Since $B_1(x_j)$ in the metric $r_j^{-2}g_j$ is exactly the ball $B_{r_j}(x_j)$ in the metric $g_j$ and $\epsilon _j\rightarrow 0$, this means that before rescaling $(\hat{v}_j^1,\ldots ,\hat{v}_j^k): B_{r_j}(x_j)\rightarrow \mathbb {R}^k$ is $L_j$-drifted $\epsilon$-splitting when j is sufficiently large, which contradicts to the inductive hypothesis that there is no matrix A such that $A\circ u$ is $L_j$-drifted $\epsilon$-splitting on $B_{r_j}(x_j)$.

Hence, this finishes the proof of Theorem 3.6 part a).