A Bakry–Émery Almost Splitting Result With Applications to the Topology of Black Holes

Abstract

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized m-Bakry–Émery Ricci curvature, in which m is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound, as well as control on the Bakry–Émery vector field, the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized m-Bakry–Émery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson’s finiteness of fundamental group isomorphism types, volume comparison, the Abresch–Gromoll inequality, and a Cheng–Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature.

A Cheng–Yau Gradient Estimate

Lemma A.1. Let (M, g, X) be a complete Riemannian manifold of dimension n with smooth vector field X. Let $m,\mathcal {C}>0$, $\delta \ge 0$, and $0<r_1<r_2$, and assume that $\mathrm {Ric}_X^m(g)\ge -(n-1)\delta g$ together with $|X|\le \mathcal {C}$ on $B_{r_2}(p)$. Suppose that $u\in C^{\infty }(B_{r_2}(p))$ is positive and satisfies

$$\begin{aligned} \Delta _X u = a F(u), \end{aligned}$$

(A.1)

for some functions $a\in C^{\infty }(B_{r_2}(p))$ and $F\in C^{\infty }(\mathbb {R}_+)$. Then there exists a constant $C_0\ge 1$ depending on $n,m,\delta ,r_1,r_2,\mathcal {C}$ such that

$$\begin{aligned} {\text {sup}}_{B_{r_1}(p)}|\nabla \log u|^2 \!\le \! C_0 +{\text {sup}}_{B_{r_2}(p)}\!\left\{ \!8n\left[ \left( |a|+|\nabla a|\right) \frac{|F(u)|}{u}\!+\! |aF'(u)| \right] \!+\! 4\!\left( \!\mathcal {C}\!+\!\sqrt{\frac{|F(u)|}{u}}\right) ^2 \!\right\} . \end{aligned}$$

(A.2)

Proof

The proof involves a detailed but straightforward calculation that appears in [7, Chapter 7], which we modify to accommodate the vector field X. Using equation (A.1) a direct computation shows that for $v:=\log u$ we obtain

$$\begin{aligned} \Delta _X v = -\left| \nabla v \right| ^2 +ae^{-v}F(e^v)=:-\left| \nabla v \right| ^2 +aG(v). \end{aligned}$$

(A.3)

Next, define $Q:=\phi |\nabla v |^2$ where the nonnegative cut-off function $\phi :B_{r_2}(p)\rightarrow [0,1]$ is chosen such that $\phi = 1$ on $B_{r_1}(p)$, $\phi =0$ in a neighborhood of $\partial B_{r_2}(p)$, and $\phi \le 1$ on $B_{r_2}(p)$. In what follows, calculations will be evaluated at a point $q\in B_{r_2}(p)$ where Q takes its maximum, so terms involving $\nabla Q$ will be dropped or rather the identity $0=|\nabla v|^2 \nabla \phi +\phi \nabla \left( |\nabla v|^2 \right)$ will be implemented.

First observe that

$$\begin{aligned} \Delta _X Q = \frac{Q}{\phi } \Delta _X \phi -\frac{2Q}{\phi ^2} |\nabla \phi |^2 +\phi \Delta _X \left( |\nabla v|^2 \right) \ . \end{aligned}$$

(A.4)

The last term in this formula may be replaced with help from the Bochner formula [31, Lemma 4]

$$\begin{aligned} \begin{aligned} \Delta _X \left( |\nabla v|^2 \right) =&\, 2|{\mathrm{Hess}}v|^2 +2\mathrm {Ric}_X^m (\nabla v, \nabla v) +2\nabla _{\nabla v}\Delta _X v +\frac{2}{m}\left( X(v)\right) ^2\\ \ge&\,\frac{2}{n}\left( \Delta v\right) ^2 -\frac{2(n-1)\delta }{\phi } Q+2\nabla _{\nabla v}\Delta _X v +\frac{2}{m}\left( X(v)\right) ^2\ , \end{aligned} \end{aligned}$$

(A.5)

where the Bakry–Émery Ricci curvature lower bound and the Cauchy–Schwarz inequality were used. Furthermore by (A.3)

$$\begin{aligned} \begin{aligned} \frac{2}{n}\phi \left( \Delta v\right) ^2=\,&\frac{2}{n}\phi ^{-1} \left( \phi \Delta _X v+\phi X(v)\right) ^2\\ =\,&\frac{2}{n}\left( \phi G(v) +\phi X(v)-Q\right) ^2, \end{aligned} \end{aligned}$$

(A.6)

and

$$\begin{aligned} 2\phi \nabla _{\nabla v}\Delta _X v =&2\phi \nabla _{\nabla v}\left( aG(v)-|\nabla v |^2 \right) \\ =&2\phi (\nabla v\cdot \nabla a)G(v)+2aG'(v)Q-2\phi \nabla v \cdot \nabla \left( |\nabla v |^2\right) \nonumber \\ =&2\phi (\nabla v\cdot \nabla a)G(v)+2aG'(v)Q+2|\nabla v |^2 \nabla v \cdot \nabla \phi \nonumber \\ =&2\phi (\nabla v\cdot \nabla a)G(v)+2aG'(v)Q+\frac{2}{\phi } Q \nabla v \cdot \nabla \phi \nonumber \\ \ge&-2|\nabla a||G(v)|\phi ^{1/2}Q^{1/2}+2aG'(v)Q-4n \frac{|\nabla \phi |^2}{\phi ^2}Q -\frac{1}{4n\phi }Q^2.\,\,\,\quad \nonumber \end{aligned}$$

(A.7)

Gathering the above expressions produces

$$\begin{aligned} \phi \Delta _X Q \ge&Q\Delta _X \phi -(2+4n) \frac{|\nabla \phi |^2}{\phi }Q -2|\nabla a||G(v)|\phi ^{3/2}Q^{1/2}+2a\phi G'(v)Q-\frac{1}{4n}Q^2 \nonumber \\&-2(n-1)\delta \phi Q +\frac{2}{m}\phi \left( X(v)\right) ^2 +\frac{2}{n}\left( \phi X(v) +\phi G(v) -Q \right) ^2 \end{aligned}$$

(A.8)

at q, where Q takes its maximum.

Now suppose that $Q(q)\le 2\phi \left( G(v)+X(v) \right) (q)$, then the definitions of v, G, and Q yield

$$\begin{aligned} |\nabla \log u|^2\le 2 u^{-1}\left( F(u)+X(u)\right) \le 2u^{-1}|F(u)| +2\mathcal {C}|\nabla \log u |\quad \quad \text {at}\quad \quad q. \end{aligned}$$

(A.9)

It follows that

$$\begin{aligned} {\text {sup}}_{B_{r_2}(p)} Q\le 4\left( \mathcal {C}+{\text {sup}}_{B_{r_2}(p)} \sqrt{u^{-1}|F(u)|}\right) ^2. \end{aligned}$$

(A.10)

If on the other hand $Q(q)\ge 2\phi \left( G(v)+X(v) \right) (q)$, then this may be manipulated into the form

$$\begin{aligned} \frac{2}{n}\left( \phi X(v) + \phi G(v) -Q\right) ^2 -\frac{1}{4n}Q^2 \ge \frac{1}{4n}Q^2 . \end{aligned}$$

(A.11)

Inserting this into (A.8) and using that $\Delta _X Q\le 0$ at the maximum point q, gives rise to

$$\begin{aligned} \frac{1}{4n}Q \le -\Delta _X \phi +(2+4n) \frac{|\nabla \phi |^2}{\phi } +2|\nabla a||G(v)|\phi ^{3/2}Q^{-1/2} -2a\phi G'(v) +2(n-1)\delta \phi . \end{aligned}$$

(A.12)

We may assume that $Q(q)> 1$, otherwise (A.2) is automatically valid since $C_0 \ge 1$. It follows that

$$\begin{aligned} {\text {sup}}_{B_{r_2}(p)}Q\le C_0 +{\text {sup}}_{B_{r_2}(p)}8n\left[ (|a|+|\nabla a|)u^{-1}|F(u)|+|aF'(u)| \right] . \end{aligned}$$

(A.13)

In order to show that the constant $C_0$ depends only on the quantities stated in the lemma, we choose the cut-off function $\phi$ to be a non-increasing function of the distance $\rho$ from p, so that as in Corollary 2.3 we have $\Delta _X\phi \ge \bar{\Delta }_{n+m}\phi$. Note that a modification employing a barrier function produces the same result when q is a cut point (see [7, page 41]).

Finally observe that the sequence of elementary inequalities

$$\begin{aligned} {\text {sup}}_{B_{r_1}(p)}|\nabla \log u|^2={\text {sup}}_{B_{r_1}(p)}Q \le {\text {sup}}_{B_{r_2}(p)}Q, \end{aligned}$$

(A.14)

together with (A.10) and (A.13) gives the desired result. $\quad \square$