The structure of spaces with Bakry–Émery Ricci curvature bounded below

Lemma B.1.

Under the conditions of Lemma 2.4, for a vector field X on Ap⁢(a,b) which satisfies

there exists an f-harmonic function θ defined in Ap⁢(a2,b2) such that

and

where Ap⁢(a3,b3) is an even smaller annulus in Ap⁢(a2,b2).

Proof.

Let h be the f-harmonic function constructed in (2.5) and let θ1=〈X,∇⁡h〉. Then ∇⁡θ1=〈∇⁡X,∇⁡h〉+〈X,hess⁢h〉. It follows that

Thus by (B.1) and Lemma 2.5, we get

Let θ be a solution of the equation

with θ=θ1 on ∂⁡Ap⁢(a2,b2). Then

It follows that

On the other hand, since

we have

By the Hölder inequality, we get

Hence,

Therefore, combining (B.1) and (B.4), we derive (B.2) immediately.

To get (B.3), we choose a cut-off function ϕ which is supported in Ap⁢(a2,b2) with bounded gradient and f-Laplace as in Lemma 1.5. Then by the Bochner identity, we have

Since

we obtain

Therefore, using integration by parts, we derive (B.3) from (B.2). ∎

Lemma B.2.

Let (M,g) be a Riemannian manifold which satisfies (3.3). Let h+ be an f-harmonic function which satisfies

Then there exists a Ψ⁢(δ;A,Λ,n)-Gromov–Hausdorff approximation from the ball Bp⁢(18) to the ball B(0×x)⁢(18)⊂R×X.

Lemma B.3.

Under condition (3.3), for an f-harmonic function h+ which satisfies (B.5)–(B.7) in Bp⁢(1), there exists a Lipschitz function ρ in Bp⁢(14) such that |h+-ρ|<Ψ and

Proof.

First, we note that the following Poincaré inequality holds for any C1-function h,

where

This is in fact a consequence of Lemma 3.4 by applying the function e to |∇⁡h|2, because

Thus by taking h=|∇⁡h+|2, we get from (B.5)–(B.7) that

Next we apply [2, Theorem 16.32] to h+ with conditions (B.5), (B.6) and (B.9). It is sufficient to check a doubling condition for the measure d⁢vf and an (ϵ,δ)-inequality. The (ϵ,δ)-inequality says that, for any ϵ,δ>0 and two points x,y∈M with d⁢(x,y)=r, there exist Cϵ,δ and another two points x′,y′ with d⁢(x′,x)≤δ⁢r and d⁢(y′,y)≤δ⁢r, respectively, such that

where

and the infimum takes among all curves from x to y with length l≤(1+ϵ)⁢d⁢(x,y). The doubling condition follows from Volume Comparison Theorem 1.2, and the (ϵ,δ)-inequality follows from Volume Comparison Theorem 1.2 and the segment inequality in Lemma 3.3. Thus we can construct a Lipschitz function ρ from h+ such that

Moreover, by [2, Lemma 8.17], we get (B.8). ∎

Proof of Lemma B.2.

As in the proof of Proposition 3.6, we define X=(h+)-1⁢(0) and the map u by

where xq is the nearest point in X to q. To show that u is a Gromov–Hausdorff approximation, we shall use Lemma 3.2. In fact, by (B.8) in Lemma B.3, we see

Then instead of (3.1) by (B.10), Lemma 3.2 is still true since (B.7) holds [3]. Hence the proof of Proposition 3.6 works for Lemma B.2. ∎

Proposition B.4.

Suppose that

Then either Bp⁢(18) is close to a Euclidean ball in the Gromov–Hausdorff topology or for a suitable choice of the orthogonal coordinates in R2⁢n-2, the map Φ=(h1,…,h2⁢n-2) constructed in Section 5 satisfies

Proof.

Roughly speaking, if the space spanned by ∇⁡hi is not almost 𝐉-invariant, we can find a vector field nearly perpendicular to these ∇⁡hi, and it satisfies condition (B.1) in Lemma B.1. Then by Lemma B.2, Bp⁢(1) will be almost split off along a new line. This implies that Bp⁢(18) is close to a Euclidean ball.

Let V be a (4⁢n-4)-dimensional line space spanned by ∇⁡hi,𝐉⁢∇⁡hi with the L2-inner product

Then 𝐉 induces an complex structure on V such that the inner product is 𝐉-invariant. We introduce a distance in the Grassmannian G⁢(2⁢n,k) as follows:

for any two k-dimensional subspaces Λ1,Λ2 in ℝ2⁢n, where the vectors ei form a unit orthogonal basis of Λ1 and prΛ2⊥ is the complement of the orthogonal projection to Λ2. First we suppose that

where

Then by the Gram–Schmidt process, one can find a unit orthogonal basis {wi} of W such that

This is equivalent to the existence of a matrix ai⁢j∈GL⁡(2⁢n-2,ℝ) which is nearly orthogonal such that

Thus by changing the orthogonal basis in ℝ2⁢n-2, (B.11) will be true.

Secondly, we suppose that

This implies that there exists some j such that

Let

Then prW⊥⁢(𝐉⁢∇⁡hi) is perpendicular to W with

and it satisfies condition (B.1) in Lemma B.1. Thus we see that there exists an f-harmonic function θ which satisfies conditions (B.5), (B.6) and (B.7) in Lemma B.2. As a consequence, Bp⁢(18) will almost spilt off along a new line associated to the coordinate function θ. Since X∈W⊥, it follows that Bp⁢(18) in fact splits off ℝ2⁢n-1 almost. But the latter implies that Bp⁢(18) is close to a Euclidean ball in the Gromov–Hausdorff topology by using a topological argument as in [6, Theorem 6.2] or by Proposition B.5 below for Kähler manifolds. ∎

Proposition B.5.

Let Y be a limit space of a sequence of Kähler manifolds in Theorem 5.1. Then

Proof.

It is sufficient to show that if a tangent cone Ty⁢Y at a point y∈Y can split off ℝ2⁢k+1, then Ty⁢Y can split off ℝ2⁢k+2. Let hi, i=1,…,2⁢k+1, be f-harmonic functions which approximate 2⁢k+1 distance functions with different directions as constructed in Section 2 and Section 3. Then as in the proof of Proposition B.4, we consider a linear space V=span⁡{∇⁡hi,𝐉⁢∇⁡hi} with L2-inner product. Since the dimension of W=span⁡{∇⁡hi} is odd, we have d⁢(W,𝐉⁢W)≥1. Thus Ty⁢Y will split off a new line. The proposition is proved. ∎