Yamabe Flow and Myers Type Theorem on Complete Manifolds

Abstract

In this paper, we prove the following Myers type theorem: If (M ⁿ,g), n≥3, is an n-dimensional complete locally conformally flat Riemannian manifold with bounded Ricci curvature satisfying the Ricci pinching condition Rc≥ϵRg, where R>0 is the scalar curvature and ϵ>0 is a uniform constant, then M ⁿ must be compact.

Appendix A

In this section, we prove Theorem 4.4 (see also [16]) for the convenience of the reader.

Proof of Theorem 4.4

We argue by contradiction. If (M ⁿ,g(t)) is the gradient expanding Yamabe soliton, then

(A.1)

at any fixed time t=t ₀>0 for some constant ρ>0. It follows that

(A.2)

Taking the trace of (A.1), we have n(R+ρ)=Δf. Substituting this into (A.2), we get

(A.3)

Take any p∈M ⁿ and assume that γ(s) is any normal geodesic with respect to g(t ₀) emanating from p. Let F(s)=f∘γ(s). By (A.1),

(A.4)

Then f is a strictly uniform convex function on M ⁿ and f has a unique minimum point. Without loss of generality, we may assume the minimum point is p and f(p)=0. Then ∇f(p)=0. By (A.3), we have −(n−1)∇ _i R∇ _i f=R _ij ∇ _j f∇ _i f. Then,

It follows from the pinching condition (4.14) that

(A.5)

where C ₁ is some positive constant depending only on the dimension n and constant ϵ. If γ(s), $0\leq s\leq\bar{s}$, is the shortest geodesic, then

by Proposition 1.94 in [9], where C ₂ is a constant depending only on the bound of curvature at t=t ₀. Then we have

(A.6)

by the pinching condition (4.14), where C ₃ is a constant depending only on the bound of curvature at t=t ₀ and the constant ϵ. From (A.1), we know that (R+ρ)=∇ _γ′∇ _γ′ f. Integrating this and using (A.6), we get

(A.7)

and

(A.8)

where C ₄ is a constant depending only on the bound of curvature at t=t ₀ and the constant ϵ. Let d(x,p) be the distance between x and p with respect to g(t ₀). It follows from (A.4) that

(A.9)

Then by (A.7), (A.8), and (A.9), we have

(A.10)

where C ₅ is a constant depending only on the bound of curvature at t=t ₀ and the constant ϵ, and C ₆ is a constant only depending on the bound of curvature at t=t ₀, the constant ϵ, and ρ. By Theorem 2.2, we see that

(A.11)

where C ₇ is a constant depending only on dimension n and the bound of curvature at t=t ₀. Then combining (A.5), (A.10), and (A.11), we obtain

(A.12)

where C ₈ is a constant depending only on the bound of curvature at t=t ₀, the dimension n, the constant ϵ, and ρ. This means that

(A.13)

where $\widetilde{C_{1}}$ is a constant depending only on the bound of curvature at t=t ₀, the dimension n, the constant ϵ, and ρ. Substituting (A.13) into the right-hand side of (A.12), we get

(A.14)

where C ₂ is a constant with the same dependence as $\widetilde{C_{1}}$. Repeating this process, we can obtain

(A.15)

where C ₅ is a constant with the same dependence as $\widetilde{C_{1}}$. Thus it follows from the result in [15] that the volume growth is Euclidean, i.e., $\lim _{r\to\infty}\frac{\operatorname {vol}(B_{r}(p))}{\mathrm{vol}_{eu}(B_{r})}=1$, where B _r (p) is the ball of radius r around p on M ⁿ. We then conclude that M ⁿ is isometric to ℝⁿ by the standard volume comparison theorem. This is a contradiction. This completes the proof. □