Integral Curvature Bounds and Bounded Diameter with Bakry--Emery Ricci Tensor

TL;DR

This paper extends classical geometric theorems to weighted Riemannian manifolds with bounded Bakry--Emery Ricci tensor and potential function, establishing compactness results under these conditions.

Contribution

It generalizes Myers' theorem to manifolds with a weighted measure and bounded Bakry--Emery Ricci tensor, providing new compactness criteria.

Findings

Proves a generalized Myers compactness theorem for weighted manifolds.

Establishes diameter bounds under Bakry--Emery Ricci curvature conditions.

Shows boundedness of the potential function f is crucial for the results.

Abstract

For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.

Equations

$$Ri{c}_f=Ric+\mathrm{Hess}f,$$

$$\mathrm{diam}(M)\le \frac{\pi }{\sqrt{H}}+\frac{4k}{(n-1)\sqrt{H}}.$$

$$\frac{1}{vol(M)}{\int }_M((n-1)-Ri{c}_{-}{)}_{+}d{v}_g<ϵ(n,\delta ),$$

$$\underset{x}{\sup }\frac{1}{vol(B(x,R))}{\int }_{B(x,R)}((n-1)-Ri{c}_{-}{)}_{+}d{v}_g<ϵ(n,k,R,\delta ),$$

$$\underset{x}{\sup }\frac{1}{vol(B(x,R))}{\int }_{B(x,R)}((n-1)-Ri{c}_{-}{)}_{+}d{v}_g<\widetilde{ϵ}(n,k,R),$$

$$\frac{1}{vo{l}_f(M)}{\int }_M((n-1)-Ri{c}_{f-}{)}_{+}{e}^{-f}d{v}_g<ϵ(n+4k,\delta ),$$

$$\underset{x}{\sup }\frac{1}{vo{l}_f(B(x,R))}{\int }_{B(x,R)}((n-1)-Ri{c}_{f-}{)}_{+}{e}^{-f}d{v}_g<ϵ(n+4k,H,R,\delta ),$$

$$\underset{x}{\sup }\frac{1}{vo{l}_f(B(x,R))}{\int }_{B(x,R)}((n-1)-Ri{c}_{f-}{)}_{+}{e}^{-f}d{v}_g<\widetilde{ϵ}(n+4k,H,R),$$

$${\int }_{A_1\times A_2}{\int }_{{\gamma }_{x,y}}h(\gamma (s))ds(e^{-f}d{v}_g{)}^2$$

$$C(n+4k,H,R)=\underset{0<\frac{s}{2}\le u\le s}{\sup }\frac{A_H^{n+4k}(s)}{A_H^{n+4k}(u)},$$

$$R\ge \sup \{d(x,y)\mid (x,y)\in (A_1\times A_2)\},$$

$$s{n}_H^{\prime \prime }+Hs{n}_H=0$$

$$m_H^n=(n-1)\frac{s{n}_H^{\prime }}{s{n}_H},$$

$$s{n}_H^2(r)m_f(r)$$

$$m_f(r)\le m_H(r)+\frac{4k}{s{n}_H(r)},$$

$$m_f(r)\le m_H(r)\left(1+\frac{4k}{n-1}\right)=m_H^{n+4k}(r).$$

$$\frac{vo{l}_f(B(p,R))}{vo{l}_f(B(p,r))}\le \frac{vo{l}_H^{n+4k}(R)}{vo{l}_H^{n+4k}(r)}.$$

$${\int }_{A_1\times A_2}{\int }_{{\gamma }_{x,y}}((n-1)-Ri{c}_{f-}{)}_{+}ds(e^{-f}d{v}_g{)}^2$$

$$\underset{(x,y)\in A_1\times A_2}{\inf }{\int }_{{\gamma }_{x,y}}((n-1)-Ri{c}_{f-}{)}_{+}ds$$

$$vo{l}_H^{n+4k}(r)={\int }_{S^{n+4k-1}}d{s}^{n+4k-1}{\int }_0^r{t}^{n+4k-1}dt,$$

$$\frac{vo{l}_H^{n+4k}(D)}{vo{l}_H^{n+4k}(r)}=\frac{D^{n+4k}}{r^{n+4k}}.$$

$$\underset{(x,y)\in (A_1\times A_2)}{\inf }{\int }_{{\gamma }_{x,y}}((n-1)-Ri{c}_{f-}{)}_{+}ds$$

$$\sum _{i=1}^{n-1}I(b{E}_i,b{E}_i)$$

$${\int }_0^L{b}^{2}Hess(f)({\gamma }^{\prime },{\gamma }^{\prime })dt={\int }_0^L{b}^2\frac{d}{dt}⟨\nabla f,{\gamma }^{\prime }⟩dt,$$

$${\int }_0^L{b}^2\frac{d}{dt}⟨\nabla f,{\gamma }^{\prime }⟩dt$$

$$\sum _{i=1}^{n-1}I(b{E}_i,b{E}_i)\le {\int }_0^L(b^{\prime }{)}^2(n-1)dt-{\int }_0^L{b}^{2}Ri{c}_f({\gamma }^{\prime },{\gamma }^{\prime })dt.$$

$$\sum _{i=1}^{n-1}I(b{E}_i,b{E}_i)$$

$$\frac{1}{1-\frac{2}{N}}<\frac{\pi +\delta }{\pi +\frac{\delta }{2}},$$

$$L=d(x,y)\ge d(p,q)-d(p,x)-d(y,q)=D\left(1-\frac{2}{N}\right).$$

Full text

1. Introduction

One of the most fundamental results in Riemannian geometry is the Myers theorem, which states that if a complete Riemannian manifold $(M,g)$ satisfies $Ric\geq(n-1)H$ with $H>0$, then $M$ is compact and ${\rm diam}(M)\leq\frac{\pi}{\sqrt{H}}$. Here, $Ric$ is the Ricci curvature of the metric $g$. This theorem has been generalized through different approaches (see [1], [2], [6], and [8]), one of which is the effort of Wei and Wylie, who proved the theorem for manifolds with a positive lower Bakry-–Emery Ricci curvature bound in [7]. A Bakry-–Emery Ricci tensor is defined as

[TABLE]

where $f$ is a smooth function on $M$ and ${\rm Hess}\,f$ is the hessian of $f$. Where $|f|\leq k$, they proved that

[TABLE]

Several works have attempted to generalize this result (for example, see [4], [5], [9], and [10]), including that of Sprouse which can be summarized in the following three theorems.

Theorem 1.1** ([6]).**

Let $(M,g)$ be a compact Riemannian manifold of nonnegative Ricci curvature. Then, for any $\delta>0$, there exists $\epsilon=\epsilon(n,\delta)$ such that if

[TABLE]

then ${\rm diam}(M)<\pi+\delta$.

Here, $dv_{g}$ is the Riemannian volume density on $M$, $Ric_{-}(x)$ is the lowest eigenvalue of the Ricci tensor $Ric(x)$, and $h_{+}(x)=\max\{h(x),0\}$ for an arbitrary function $h$ on $M$. For $Ric\geq(n-1)k$ with $k\leq 0$, these generalizations are attained.

Theorem 1.2** ([6]).**

Let $(M,g)$ be a complete Riemannian manifold with $Ric\geq(n-1)k$, $k\leq 0$. Then, for any $R,\delta>0$, there exists $\epsilon=\epsilon(n,k,R,\delta)$ such that if

[TABLE]

then $(M,g)$ is compact and ${\rm diam}(M)<\pi+\delta$.

Theorem 1.3** ([6]).**

Let $(M,g)$ be a complete Riemannian manifold with $Ric\geq(n-1)k$, $k\leq 0$. Then, for any $R>0$, there exists $\tilde{\epsilon}=\tilde{\epsilon}(n,k,R)$ such that if

[TABLE]

then the universal cover of $M$ is compact, and hence, $\pi_{1}(M)$ is finite.

We will generalize these results to the Bakry-–Emery Ricci tensor bounded from below. Let $(M,g,e^{-f}dv_{g})$ be a smooth metric measure space, where $M$ is a complete $n$-dimensional Riemannian manifold with metric $g$. Likewise, let $Ric_{f-}(x)$ denote the lowest eigenvalue of the Bakry-–Emery Ricci tensor $Ric_{f}(x)$. Then, we prove the following theorem.

Theorem 1.4**.**

Let $(M,g,e^{-f}dv_{g})$ be a compact $n$-dimensional Riemannian manifold with $Ric_{f}\geq 0$ and $|f|\leq k$. Then, for any $\delta>0$, there exists $\epsilon=\epsilon(n+4k,\delta)$ such that if

[TABLE]

then ${\rm diam}(M)<\pi+\delta$.

Given that $(M,g)$ is noncompact or does not exhibit a nonnegative Bakry–-Emery Ricci curvature, averaging the bad part of $Ric_{f}$ over metric ball, as in [6] yields a similar result as follows.

Theorem 1.5**.**

Let $(M,g,e^{-f}dv_{g})$ be a complete $n$-dimensional Riemannian manifold with $Ric_{f}\geq(n-1)H$, $H<0$, and $|f|\leq k$. Then, for any $R,\delta>0$, there exists $\epsilon=\epsilon(n+4k,H,R,\delta)$ such that if

[TABLE]

then $M$ is compact and ${\rm diam}(M)<\pi+\delta$.

Finally, we could obtain the result for the fundamental group of $M$, which is stated as follows.

Theorem 1.6**.**

Let $(M,g,e^{-f}dv_{g})$ be a complete $n$-dimensional Riemannian manifold with $Ric_{f}\geq(n-1)H$, $H<0$, and $|f|\leq k$. Then for any $R>0$, there exists $\tilde{\epsilon}=\tilde{\epsilon}(n+4k,H,R)$ such that if

[TABLE]

then the universal cover of $M$ is compact, and hence, $\pi_{1}(M)$ is finite.

2. Proof of Theorem 1.4

Let $(M,g,e^{-f}dv_{g})$ be a smooth metric measure space, where $(M,g)$ is a complete $n$-dimensional Riemannian manifold. Let $A_{1},A_{2},W$ be open subsets of $M$ such that $A_{1},A_{2}\subset W$, and all minimal geodesics $\gamma_{x,y}$ from $x\in A_{1}$ to $y\in A_{2}$ lie in $W$.

We will use the estimate of Cheeger and Colding for Bakry–Emery Ricci tensor ([3], Proposition 2.3); thus, for a nonnegative integrable function on $M$,

[TABLE]

then,

[TABLE]

and

[TABLE]

where $A_{H}^{n+4k}(r)$ denotes the area element on $\partial B(r)$ in $M_{H}^{n+4k}$, the simply connected model space of dimension $n+4k$ with constant curvature $H$. Because $H=0$, we denote $C(n+4k,H,R)$ by $C(n+4k)$.

Applying $sn_{H}(r)$ as a solution to

[TABLE]

, $sn_{H}(0)=0$ and $sn_{H}^{\prime}(0)=1$ are satisfied. Moreover, if $H=0$, then

[TABLE]

with the solution $sn_{H}(r)=r$.

By the mean curvature comparison (3.15) in the proof of Theorem 1.1 in [7], we have

[TABLE]

Thus,

[TABLE]

implying that

[TABLE]

Therefore,

[TABLE]

Now, let $p,q\in M$ such that $d(p,q)={\rm diam}(M)=D$, $r>0$, $A_{1}=B(p,r)$, and $A_{2}=B(q,r)$. Applying the inequality (2.1),

[TABLE]

Consequently, let $vol_{H}^{n+4k}(r)$ be the volume of the radius $r$-ball in $M_{H}^{n+4k}$, the simply connected model space of dimension $n+4k$ with constant curvature $H$. Then, we have

[TABLE]

where the last inequality follows from (2.3).

Note that if $H=0$, the volume element $v(r)=r^{n+4k-1}$, which gives

[TABLE]

to obtain

[TABLE]

Therefore,

[TABLE]

Now, we can find a minimizing unit speed geodesic $\gamma$ from $x\in\overline{A_{1}}$ to $y\in\overline{A_{2}}$ of length $L=d(x,y)$. Let $\{E_{1},\cdots,E_{n}=\gamma^{\prime}\}$ be a parallel orthonormal frame along $\gamma$ and a smooth function $b\in C^{\infty}([0,L])$ such that $b(0)=b(L)=0$; then, by the second variation of $\gamma$, we have

[TABLE]

Likewise, note that

[TABLE]

thus,

[TABLE]

such that,

[TABLE]

If we set the function $b$ as $b(t)=\sin\left(\frac{\pi t}{L}\right)$, then we obtain $(b^{\prime}(t))^{2}=\frac{\pi^{2}}{L^{2}}\cos^{2}(\frac{\pi t}{L})$ and $b^{2}(t)=\sin^{2}\left(\frac{\pi t}{L}\right)$. Thus,

[TABLE]

By the inequality (2.4),

[TABLE]

Now, let $r=\frac{D}{N}$, and choose $N=N(\delta)$ such that

[TABLE]

by the triangle inequality,

[TABLE]

Thus,

[TABLE]

Setting

[TABLE]

and if

[TABLE]

then,

[TABLE]

Because $\gamma$ is a minimal geodesic such that

[TABLE]

Then by (2.7), we obtain

[TABLE]

This inequality gives

[TABLE]

Finally, by the inequality (2.5), (2.6), and (2.8), we have

[TABLE]

This completes the proof.

3. Proof of Theorem 1.5 and 1.6

We will prove Theorem 1.5 in this section following the same setting for Theorem 1.4. Let $\gamma$ be a minimizing unit speed geodesic from $x\in\overline{A_{1}}$ to $y\in\overline{A_{2}}$ of length $L=d(x,y)$. Likewise, let $\{E_{1},\cdots,E_{n}=\gamma^{\prime}\}$ be a parallel orthonormal frame along $\gamma$ and a smooth function $b\in C^{\infty}([0,L])$ such that $b(0)=b(L)=0$. For the proof, we need the following result.

Lemma 3.1**.**

Let $(M,g,e^{-f}dv_{g})$ be a complete Riemannian manifold with $Ric_{f}\geq(n-1)H$, $H<0$, and $|f|\leq k$. Then, for any fixed $R>\pi$, there exists $\epsilon=\epsilon(n+4k,H,R,\delta)$ such that if

[TABLE]

for some $B(p,R)\subset M$, then $M=B(p,R)\subset B(p,\pi+\delta)$.

Proof of Lemma 3.1. Let $sn_{H}(r)$ be the solution to

[TABLE]

such that $sn_{H}(0)=0$ and $sn_{H}^{\prime}(0)=1,$ then

[TABLE]

When $H<0$, this solution is given by $sn_{H}(r)=\frac{1}{\sqrt{-H}}\sinh(\sqrt{-H}r)$. By the inequality (2.2),

[TABLE]

thus,

[TABLE]

If we set $p\in M$ and $W=B(p,R)$, then $q$ will be any point in $W$ satisfying $\pi+4r<d(p,q)<R-3r$, where $0<r<\frac{1}{8}(R-\pi)$ is to be determined, and $A_{1}=B(p,r),A_{2}=B(q,r)$. Thus, by (3.1)

[TABLE]

Let $r=\frac{1}{4}\delta$, where $\delta<\frac{1}{2}(R-\pi)$. Then

[TABLE]

By the second variation of $\gamma$,

[TABLE]

Setting

[TABLE]

we obtain

[TABLE]

Moreover, by the minimality of $\gamma$, we have

[TABLE]

implying that

[TABLE]

By the triangle inequality,

[TABLE]

We assumed that $\pi+4r<d(p,q)<R-3r$, or $\pi+\delta<d(p,q)<R-3r$. However, by (3.2), no geodesic starting from $p$ of a length greater than $\pi+\delta$ can be length minimizing, which implies that $B(p,R)\subset B(p,\pi+\delta)$. If $R$ goes to infinity, then $B(p,R)$ tends to $M$. Hence, we may conclude that $M=B(p,R)\subset B(p,\pi+\delta)$.

Now we can prove Theorem 1.5.

Proof of Theorem 1.5. Note that Lemma 3.1 shows Theorem 1.5 for $R>\pi$. Thus, it suffices to prove the case when $R\leq\pi$.

Let $R^{\prime}>\pi$ be fixed. Then for any $R\leq\pi$, there exists $N=N(H,R,R^{\prime})>0$, such that any $R^{\prime}$-ball in $M$ can be covered by $N$ or fewer $R$-balls, $B(x_{i},R)$, $1\leq i\leq N$. Subsequently,

[TABLE]

Hence, we can conclude that

[TABLE]

Finally, let us prove Theorem 1.6.

Proof of Theorem 1.6. Let $(\tilde{M},\tilde{g})$ be a Riemannian universal cover of $(M,g)$. Because the inequality

[TABLE]

holds on $(M,g)$, the same inequality holds on $(\tilde{M},\tilde{g})$.

Based on this, it is easy to see that Theorem 1.5 with $\delta=0$ also holds. When $R\leq\pi$, we just need to follow the proof of Theorem 1.5; moreover, when $R>\pi$, setting $\pi<d(p,q)<R$ with $r=\frac{\pi}{4}$, we can prove that ${\rm diam}\,(M)\leq\pi$. Hence, we can conclude that $\tilde{M}$ is compact, implying that the fundamental group $\pi_{1}(M)$ is finite.