Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

Abstract

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry’s sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $\mathop {\mathrm {Ric}}\nolimits \ge -Kg$ and $\mathop {\mathrm {diam}}\nolimits \le D$ for some positive constants $K>0$ and $D>0$.

Appendices

Appendix A. Spherical multi-suspension

In this appendix we show the following multi-suspension theorem, which gives an approximation of the shape of the Riemannian manifold under the pinching condition on $\lambda _k({\bar{\Delta }}^E)$, based on the methods of [22]. As a consequence, we give another Proof of Main Theorem 1.

Theorem A.1

Given integers $n\ge 2$ and $1\le k\le n+1$, and positive real numbers $\epsilon >0$, $K>0$ and $D>0$, there exists $\delta (n,K,D,\epsilon )>0$ such that if (M, g) is an n-dimensional closed Riemannian manifold with $\mathop {\mathrm {Ric}}\nolimits \ge -K g$, $\mathop {\mathrm {diam}}\nolimits (M)\le D$ and $\lambda _k({\bar{\Delta }}^E)\le \delta$, then we have one of the following:

• (i) $d_{GH}(M,S^{k-1})\le \epsilon$,

• (ii) $d_{GH}(M,S^k)\le \epsilon$,

• (iii) There exists a compact geodesic space Z such that $d_{GH}(M,S^{k-1}*Z)\le \epsilon$, where $S^{k-1}*Z$ denotes the k-fold spherical suspension of Z (see the definition below).

In the following, we show that each $B_{ij}$ in Proposition 2.19 is close to the spherical suspension, and prove Theorem A.1 by iteration.

We first recall some basic definitions.

Definition A.2

Let Z be a metric space.

• (i) We say Z is a geodesic space if for each $z_1,z_2\in Z$, there exists a minimal geodesic $c:[0,d(z_1,z_2)]\rightarrow Z$ connecting them, i.e., $c(0)=z_1$, $c(d(z_1,z_2))=z_2$ and $d(c(t_1),c(t_2))=|t_1-t_2|$ holds for all $t_1,t_2\in [0,d(z_1,z_2)]$. Given points $z_1,z_2\in Z$, $\gamma _{z_1,z_2}$ denotes one of minimal geodesics connecting them.

• (ii) We define an equivalence relation $\sim$ on $[0,\pi ]\times Z$ by $(0,z_1)\sim (0,z_2)$ and $(\pi ,z_1)\sim (\pi ,z_2)$ for all $z_1,z_2\in Z$. Define a metric d on $([0,\pi ]\times Z)/\sim$ to be

  $$\begin{aligned} \cos d([t_1,z_1],[t_2,z_2])=\cos t_1\cos t_2+\sin t_1\sin t_2\cos \min \{d(z_1,z_2),\pi \} \end{aligned}$$

  and $0\le d([t_1,z_1],[t_2,z_2])\le \pi$ for all $[t_1,z_1],[t_2,z_2]\in ([0,\pi ]\times Z)/\sim$. The metric space $(([0,\pi ]\times Z)/\sim ,d)$ is called the spherical suspension of Z and denoted by $S^0*Z$. Define $0^*:= [0,z]\in S^0*Z$ and $\pi ^*:=[\pi ,z]\in S^0*Z$.

• (iii) For any positive integer $k\in {\mathbb {Z}}_{>0}$, we define $S^k*Z:=S^0*(S^{k-1}*Z)$ and $[t_1,\ldots ,t_{k+1},z]:=[t_1,[t_2,\ldots , t_{k+1},z]]\in S^k*Z$ inductively. We call $S^k*Z$ the k-fold spherical suspension of Z.

• (iv) We define an equivalence relation $\sim '$ on $[0,\infty ) \times Z$ by $(0,z_1)\sim ' (0,z_2)$ for all $z_1,z_2\in Z$. Define a metric d on $([0,\infty )\times Z)/\sim '$ by

  $$\begin{aligned} d([t_1,z_1],[t_2,z_2]):=\left( t_1^2+t_2^2-2t_1 t_2 \cos \min \{d(z_1,z_2),\pi \}\right) ^\frac{1}{2}. \end{aligned}$$

  The metric space $(([0,\infty )\times Z)/\sim ',d)$ is called the metric cone of Z and denoted by C(Z). Put $0^*=[0,z]\in C(Z)$.

The spherical suspension is the generalization of the warped product metric with warping function $\sin t$. For an $(n-1)$-dimensional Riemannian manifold $(Z,g_{n-1})$, the distance on $(0,\pi )\times Z$ induced by the warped product metric $g=d t^2+ \sin ^2 t g_{n-1}$ coincides with the spherical suspension metric. If the Riemannian metric extends smoothly to the endpoints $0^*$ and $\pi ^*$ in $([0,\pi ]\times Z)/\sim$, then $(Z,g_{n-1})$ is isometric to the $(n-1)$-dimensional standard sphere of radius 1. Note that the warped product metric on $([0,\pi ]\times S^{n-1})/\sim$ with warping function $\sin t$ coincides with the metric on the n-dimensional standard sphere, and so we have $S^n=S^0*S^{n-1}$. Inductively we have $S^n=S^{n-1}*\{0,\pi \}$.

Definition A.3

Let (X, x, d) be a pointed metric space.

• (i) We say a sequence of pointed metric spaces $\{(X_i,x_i,d_i)\}_{i\in {\mathbb {N}}}$ converges to (X, x, d) in the pointed Gromov–Hausdorff sense if for all $R>0$, there exists a sequence of positive real numbers $\{\epsilon _i\}_{i\in {\mathbb {N}}}$ such that $\lim _{i\rightarrow \infty }\epsilon _i= 0$ and a sequence of $\epsilon _i$-Hausdorff approximation maps $\{\psi _i:B_R(x)\rightarrow B_R(x_i)\}_{i\in {\mathbb {N}}}$.

• (ii) We say a pointed metric space $(Y,y,d')$ is the tangent cone of X at x if there exists a sequence of positive real numbers $\{r_i\}_{i\in {\mathbb {N}}}$ such that $\lim _{i\rightarrow \infty }r_i= 0$ and $\{(X,x,r_i^{-1} d)\}_{i\in {\mathbb {N}}}$ converges to $(Y,y,d')$ in the pointed Gromov–Hausdorff sense.

Note that if a (pointed) metric space X (or (X, x)) is a (pointed) Gromov–Hausdorff limit of a sequence of proper geodesic spaces, then X is also a proper geodesic space.

We next recall some facts and definitions about Ricci limit spaces. Let $\{(M_i,g_i,p_i)\}$ be a sequence of pointed closed Riemannian manifolds with $\mathop {\mathrm {Ric}}\nolimits _i\ge -K g_i$ and $\mathop {\mathrm {diam}}\nolimits (M_i)\le D$ ($K,D>0$). Then, there exist a subsequence (denote it again by $\{(M_i,g_i,p_i)\}$) and a pointed geodesic space $(X,p,\nu )$ with a Radon measure $\nu$ such that $\{(M_i,g_i)\}$ converges to (X, p) in the pointed Gromov–Hausdorff sense and

$$\begin{aligned} \lim _{i\rightarrow \infty } \frac{\mathop {\mathrm {Vol}}\nolimits (B_r(x_i))}{\mathop {\mathrm {Vol}}\nolimits (B_1(p_i))}=\nu (B_r(x)) \end{aligned}$$

for all $r>0$, $x_i\in M_i$ and $x\in X$ with $\lim _{i\rightarrow \infty } d(\psi _i(x_i),x)=0$, where $\psi _i$ is a Hausdorff approximation map that we used for the definition of the pointed Gromov–Hausdorff convergence (see Theorem 1.6 and Theorem 1.10 of [8]). We call such $\nu$ a limit measure. We can consider the cotangent bundle $\pi :T^*X \rightarrow X$ with a canonical inner product by [5, 9] (see also [24, Section 2] for a short review). We have $\nu (X{\setminus } \pi (T^*X))=0$ and $T^*_x X:=\pi ^{-1}(x)$ is a finite dimensional vector space with an inner product for all $x\in \pi (T^*X)$. For all Lipschitz function f on X, we can define $d f(x)\in T_x^*X$ for almost all $x\in X$, and we have $d f\in L^\infty (T^*X)$. Let $\mathop {\mathrm {LIP}}\nolimits (X)$ be the set of the Lipschitz functions on X. For all $f\in \mathop {\mathrm {LIP}}\nolimits (X)$, we define $\Vert f\Vert _{H^{1,2}}^2=\Vert f\Vert _2^2+\Vert d f\Vert _2^2$. Let $H^{1,2}(X)$ be the completion of $\mathop {\mathrm {LIP}}\nolimits (X)$ with respect to this norm. Define

$$\begin{aligned} \begin{aligned}&{\mathcal {D}}^2(\Delta ,\nu ):=\Big \{f\in H^{1,2}(X) : \text {there exists }F\in L^2(X) \text { such that}\\&\quad \int _X \langle df, dh \rangle \,d \nu =\int _X F h\,d \nu \hbox { for all}\ h\in H^{1,2}(X) \Big \}. \end{aligned} \end{aligned}$$

For any $f\in {\mathcal {D}}^2(\Delta ,\nu )$, the function $F\in L^2(X)$ is uniquely determined. Thus, we define $\Delta f:=F$. For all sequence $f_i\in L^2(M_i)$ and $f\in L^2(X)$, we say that $f_i$ converges to f weakly in $L^2$ (see [24]) if

$$\begin{aligned} \sup _{i\in {\mathbb {N}}}\Vert f_i\Vert _2<\infty , \end{aligned}$$

and for all $r>0$, $x_i\in M_i$ and $x\in X$ with $\lim _{i\rightarrow \infty }d(\psi _i(x_i),x)=0$, we have

$$\begin{aligned} \lim _{i\rightarrow \infty }\frac{1}{\mathop {\mathrm {Vol}}\nolimits (B_1(p_i))}\int _{B_r(x_i)} f_i \,d \mu _{g_i}=\int _{B_r(x)} f \,d \nu . \end{aligned}$$

We say that $f_i$ converges to f strongly in $L^2$ if $f_i$ converges to f weakly in $L^2$, and

$$\begin{aligned} \limsup _{i\rightarrow \infty } \Vert f_i\Vert _2\le \Vert f\Vert _2 \end{aligned}$$

holds. Note that if $f_i$ converges to f weakly in $L^2$, then

$$\begin{aligned} \liminf _{i\rightarrow \infty }\Vert f_i\Vert _2\ge \Vert f\Vert _2 \end{aligned}$$

by [24, Proposition 3.29].

We need the following easy lemma.

Lemma A.4

Let (M, g) be a complete Riemannian manifold. Take arbitrary $\epsilon >0$. If points $p,a,b\in M$ satisfies $d(p,a)+d(a,b)\le d(p,b)+\epsilon$ and $\gamma _{a,b}(s)\in I_p{\setminus } \{p\}$ for almost all $s\in [0,d(a,b)]$, then we have $d(p,a)+d(a,\gamma _{a,b}(s))\le d(p,\gamma _{a,b}(s))+\epsilon$ for all $s\in [0,d(a,b)]$, and

$$\begin{aligned} \int _0^{d(a,b)}\left( 1-\langle {\dot{\gamma }}_{a,b}(s),\frac{\partial }{\partial r}\rangle \right) \, d s \le \epsilon , \end{aligned}$$

where $\frac{\partial }{\partial r}$ denotes ${\dot{\gamma }}_{p,x}(d(p,x))\in T_x M$ for all $x\in I_p{\setminus } \{p\}$.

Remark A.1

Since $\left| {\dot{\gamma }}_{a,b}(s)-\frac{\partial }{\partial r}\right| ^2 =2\left( 1-\langle {\dot{\gamma }}_{a,b}(s),\frac{\partial }{\partial r}\rangle \right)$, we have

$$\begin{aligned} \int _0^{d(a,b)}\left| {\dot{\gamma }}_{a,b}(s)-\frac{\partial }{\partial r}\right| \, d s \le (2 d(a,b)\epsilon )^{\frac{1}{2}}. \end{aligned}$$

Proof

The first assertion is trivial. The map $d(\gamma _{a,b}(\cdot ),p):[0,d(a,b)]\rightarrow {\mathbb {R}}_{>0}$ is Lipschitz continuous and $(d(\gamma _{a,b}(s),p))'=\langle {\dot{\gamma }}_{a,b}(s),{\partial }/{\partial r}\rangle$ for all s with $\gamma _{a,b}(s)\in I_p{\setminus } \{p\}$. Since

$$\begin{aligned} \int _0^{d(a,b)} (d(\gamma _{a,b}(s),p))' \,d s=d(b,p)-d(a,p), \end{aligned}$$

we get the lemma. $\square$

The following proposition asserts that each $B_{ij}$ in Proposition 2.19 is close to the spherical suspension.

Proposition A.5

Given an integers $n\ge 2$, and positive real numbers $K>0$ and $D>0$, then there exists $\eta (n,K,D)>0$ such that the following property holds. Take a positive real number $0<\delta \le \eta$. Let (M, g) be an n-dimensional closed Riemannian manifold with $\mathop {\mathrm {Ric}}\nolimits \ge -Kg$ and $\mathop {\mathrm {diam}}\nolimits (M)\le D$. Suppose that a non-zero function $f\in C^\infty (M)$ satisfies $\Vert \nabla ^2 f+f g\Vert _2\le \delta \Vert f\Vert _2$. Under the notation of Proposition 2.19, for all i, j with $|d(x_i,x_j)-\pi |\le \delta ^\frac{1}{200n^2}$, we have $d_{GH}(B_{ij},S^0*Z_{ij})\le C(n,K,D) \delta ^{\frac{1}{4000n^2}}$, where we put $Z_{ij}:=\{x\in B_{ij}:d(x_i,x)=d(x_j,x)\}$.

Proof

We first suppose that $\delta \le \delta _{\mathop {2}\nolimits }$. Since we have either $m_i=m_j+1$ or $m_i=m_j-1$, we can assume that $x_i\in A_{m_i}$ and $x_j\in A_{m_i+1}$.

Let us apply Lemma 2.14 to each $y_1\in Q$ and $y_2\in D(y_1)$.

Claim A.6

For all $y_1\in Q$ and $y_2\in D(y_1)$ (see Lemma 2.12), we have

$$\begin{aligned} \begin{aligned} \big |\cos d(p,y_1) -&\cos d(p,y_2) \cos d(y_1,y_2) \\&+\langle \nabla f_1 (y_2),{\dot{\gamma }}_{y_1,y_2}\rangle \sin d(y_1,y_2)\big |\le C(n,K,D)\delta ^{\frac{1}{48n}}. \end{aligned} \end{aligned}$$

Proof of Claim A.6

Take arbitrary $y_1\in Q$ and $y_2\in D(y_1)$. Since we have

$$\begin{aligned} \int _0^{d(y_1,y_2)} \left| \frac{\partial ^2}{\partial s^2} (f_1\circ \gamma _{y_1,y_2}(s)) + f_1 \circ \gamma _{y_1,y_2}(s)\right| \, d s \le \delta ^{\frac{1}{6}}, \end{aligned}$$

we get $\left| f_1\circ \gamma _{y_1,y_2}(d(y_1,y_2)-s) - f_1(y_2) \cos s + \langle \nabla f_1 (y_2),{\dot{\gamma }}_{y_1,y_2}\rangle \sin s\right| \le \sinh D \delta ^{\frac{1}{6}}$ for all $s\in [0,d(y_1,y_2)]$ by applying Lemma 2.14 to $f_1\circ \gamma _{y_1,y_2}(d(y_1,y_2)-s)$. Putting $s=d(y_1,y_2)$, we get the claim by $\Vert f_1-\cos d(p,\cdot )\Vert _{\infty }\le C\delta ^{\frac{1}{48n}}$ (see Proposition 2.15). $\square$

Let us replace $\nabla f$ by $\nabla \cos d(p,\cdot )$ in Claim A.6 using $\Vert \nabla f_1-\nabla \cos d(p,\cdot )\Vert _2\le C\delta ^{\frac{1}{48n}}$ (see Proposition 2.15), and get the integral pinching condition.

Claim A.7

For all $y_1\in Q$, we define $F_{y_1}:M\rightarrow {\mathbb {R}}_{\ge 0}$ by

$$\begin{aligned} F_{y_1}(y_2):=\Big |\cos d(p,y_1) - \cos d(p,y_2) \cos d(y_1,y_2) - \langle {\dot{\gamma }}_{y_1,y_2},\frac{\partial }{\partial r} \rangle \sin d(p,y_2) \sin d(y_1,y_2)\Big | \end{aligned}$$

for all $y_2 \in I_p\cap I_{y_1}{\setminus }\{p,y_1\}$ and $F_{y_1}(y_2)=0$ otherwise, where $\frac{\partial }{\partial r}$ denotes ${\dot{\gamma }}_{p,x}(d(p,x))\in T_x M$ for all $x\in I_p{\setminus } \{p\}$. Then, we have

$$\begin{aligned} \frac{1}{\mathop {\mathrm {Vol}}\nolimits (M)}\int _M F_{y_1} \,d\mu _g\le C(n,K,D)\delta ^{\frac{1}{48n}}. \end{aligned}$$

Proof of Claim A.7

Take arbitrary $y_1\in Q$. Since have $\mathop {\mathrm {Vol}}\nolimits (M{\setminus } D(y_1))\le \delta ^\frac{1}{6}\mathop {\mathrm {Vol}}\nolimits (M)$ and $F_{y_1}\le 3$, we get

$$\begin{aligned} \begin{aligned}&\frac{1}{\mathop {\mathrm {Vol}}\nolimits (M)}\int _M F_{y_1} \,d\mu _g \le \frac{1}{\mathop {\mathrm {Vol}}\nolimits (M)}\int _{D(y_1)} \Big |\cos d(p,y_1) - \cos d(p,y_2) \cos d(y_1,y_2)\\&\qquad + \langle \nabla f_1 (y_2),{\dot{\gamma }}_{y_1,y_2} \rangle \sin d(y_1,y_2)\Big |\, d y_2\\&\qquad +\Vert \nabla f_1+\sin d(p,\cdot )\frac{\partial }{\partial r}\Vert _1+3 \delta ^\frac{1}{6}\\&\quad \le C\delta ^{\frac{1}{48n}} \end{aligned} \end{aligned}$$

by Claim A.6 and Proposition 2.15. Thus, we get the claim. $\square$

Similarly to Lemma 2.12, let us get the pinching condition on the segments using the integral pinching condition on the manifold.

Claim A.8

For each $y_1\in Q$ and $y_2\in M$, we define

$$\begin{aligned} \begin{aligned} E_{y_1}(y_2):=\Big \{y_3\in I_{y_2}&{\setminus }\{y_2\}: \int _0^{d(y_2,y_3)} F_{y_1} \circ \gamma _{y_2,y_3}(s)\, d s\le \delta ^\frac{1}{144n},\text { and we have} \\&\gamma _{y_2,y_3}(s)\in I_p\cap I_{y_1}{\setminus } \{p,y_1\} \hbox { for almost all}\ s\in [0,d(y_2,y_3)] \Big \}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} R_{y_1}:=\{y_2\in M: \mathop {\mathrm {Vol}}\nolimits (E_{y_1}(y_2))\ge (1-\delta ^\frac{1}{144n})\mathop {\mathrm {Vol}}\nolimits (M)\}. \end{aligned}$$

Then, we have

$$\begin{aligned} \mathop {\mathrm {Vol}}\nolimits (R_{y_1})\ge (1- C(n,K,D)\delta ^\frac{1}{144n})\mathop {\mathrm {Vol}}\nolimits (M). \end{aligned}$$

Proof of Claim A.8

By applying the segment inequality Theorem 2.11 to functions $F_{y_1}$ and $1-\chi _{I_p\cap I_{y_1}{\setminus } \{p,y_1\}}$ (here $\chi _{I_p\cap I_{y_1}{\setminus } \{p,y_1\}}(x)=1$ for all $x\in I_p \cap I_{y_1}{\setminus } \{p,y_1\}$ and $\chi _{I_p\cap I_{y_1}{\setminus } \{p,y_1\}}(x)=0$ otherwise), we get the claim by Claim A.7 similarly to Lemma 2.12. $\square$

Now, we define an approximation map. Define $\phi _{ij}:B_{ij}\rightarrow Z_{ij}$ as follows. If $x\in B_{ij}$ satisfies $d(x,x_i)\le d(x,x_j)$, then

$$\begin{aligned} d(x_i,\gamma _{x,x_j}(0))&\le d(x_j,\gamma _{x,x_j}(0)),\\ d(x_i,\gamma _{x,x_j}(d(x,x_j)))\ge&d(x_j,\gamma _{x,x_j}(d(x,x_j)))=0. \end{aligned}$$

Thus, we can choose $s\in [0,d(x,x_j)]$ with $d(x_i,\gamma _{x,x_j}(s))= d(x_j,\gamma _{x,x_j}(s))$, and define $\phi _{ij}(x):=\gamma _{x,x_j}(s)\in \mathop {\mathrm {Im}}\nolimits \gamma _{x,x_j}\cap Z_{ij}$. Similarly, if $d(x,x_i)> d(x,x_j)$, we define $\phi _{ij}(x)$ to be $\phi _{ij}(x)\in \mathop {\mathrm {Im}}\nolimits \gamma _{x,x_i}\cap Z_{ij}$. We define an approximation map $\psi _{ij}:B_{ij}\rightarrow S^0*Z_{ij}$ by

$$\begin{aligned} \psi _{i j} (x) :=\left\{ \begin{array}{cc} 0^*&{}\quad (d(x,x_i)\le \delta ^\frac{1}{2000n^2}),\\ {[}d(x,x_i),\phi _{ij}(x)]&{} \quad (d(x,x_i)>\delta ^\frac{1}{2000n^2} \text { and } d(x,x_j)>\delta ^\frac{1}{2000n^2}),\\ \pi ^*&{}\quad (d(x,x_j)\le \delta ^\frac{1}{2000n^2}). \end{array}\right. \end{aligned}$$

We list some basic properties of the metric on $B_{i j}$.

Claim A.9

We have the following properties.

• (i) For all $x\in Z_{ij}$, we have $|d(x,x_i)-\pi /2|\le C\delta ^\frac{1}{250n^2}$ and $|d(x,x_j)-\pi /2|\le C\delta ^\frac{1}{250n^2}$.

• (ii) For all $x\in B_{ij}$, we have $|d(p,x)-(m_i \pi +d(x,x_i))|\le C\delta ^\frac{1}{250n^2}$ and $|d(p,x)-((m_i +1)\pi -d(x,x_j))|\le C\delta ^\frac{1}{250n^2}$.

• (iii) For all $x\in B_{ij}$ with $d(x,x_i)\le d(x,x_j)$, we have $d(p,x)+d(x,\phi _{ij}(x))\le d(p,\phi _{ij}(x))+ C\delta ^\frac{1}{250n^2}$.

• (iv) For all $x\in B_{ij}$ with $d(x,x_i)> d(x,x_j)$, we have $d(p,\phi _{ij}(x))+d(\phi _{ij}(x),x)\le d(p,x)+ C\delta ^\frac{1}{250n^2}$.

Claim A.9 is easy consequence of the inequalities $|d(p,x_i)-m_i\pi |\le \delta ^\frac{1}{100n}$, $|d(p,x_j)-(m_i+1)\pi |\le \delta ^\frac{1}{100n}$, $|d(x_i,x_j)-\pi |\le \delta ^\frac{1}{200n^2}$ and $d(x,x_i)+d(x,x_j)\le d(x_i,x_j)+\delta ^\frac{1}{250n^2}$ ($x\in B_{i j}$) in Proposition 2.19.

Let us compare the cosine of the metric on $B_{i j}$ and that of the spherical suspension metric (see Definition A.2 (ii)).

Claim A.10

Define $G_{i j}:=\{x\in B_{i j}: d(x,x_i)>\delta ^\frac{1}{2000n^2}, d(x,x_j)>\delta ^\frac{1}{2000n^2}\}$. Then, we have

$$\begin{aligned} \begin{aligned} \Big |\cos d(y_1,y_2)&-\cos d(x_i,y_1)\cos d(x_i,y_2) \\&-\sin d(x_i,y_1)\sin d(x_i,y_2)\cos d(\phi _{i j}(y_1),\phi _{i j}(y_2)) \Big |\le C(n,K,D)\delta ^\frac{1}{1000n^2} \end{aligned} \end{aligned}$$

for all $y_1,y_2\in G_{ij}$.

Proof of Claim A.10

Take arbitrary $y_1, y_2\in G_{i j}$. Put $y_3:=\phi _{ij}(y_2)$. By Lemma 2.12, Claim A.8 and the Bishop–Gromov inequality (Theorem 2.13), there exists points ${\tilde{y}}_1\in Q$, ${\tilde{y}}_2\in R_{{\tilde{y}}_1}$ and ${\tilde{y}}_3 \in E_{{\tilde{y}}_1}({\tilde{y}}_2)$ with $d(y_k,{\tilde{y}}_k)\le C\delta ^\frac{1}{144n^2}$ for $k=1,2,3$. Put $l(s)=d({\tilde{y}}_1,\gamma _{{\tilde{y}}_2,{\tilde{y}}_3}(s))$. Then, by the first variation formula, we have $l'(s)=\langle {\dot{\gamma }}_{{\tilde{y}}_1,\gamma _{\tilde{y_2},{\tilde{y}}_3}(s)}(l(s)), {\dot{\gamma }}_{\tilde{y_2},{\tilde{y}}_3}(s)\rangle$ for all $s\in I_{\tilde{y_1}}{\setminus } \{\tilde{y_1}\}$. Suppose that $d(y_2,x_i)\le d(y_2,x_j)$. Then, we have $d(p,{\tilde{y}}_2)+d({\tilde{y}}_2,{\tilde{y}}_3)\le d(p,{\tilde{y}}_3)+C\delta ^\frac{1}{250n^2}$ by Claim A.9 (iii). Thus, by Lemma A.4 and the definition of $E_{{\tilde{y}}_1}({\tilde{y}}_2)$, we get

$$\begin{aligned} \begin{aligned}&\int _0^{d({\tilde{y}}_2,{\tilde{y}}_3)} |\cos d(p,\tilde{y_1})-\cos (d(p,{\tilde{y}}_2)+s)\cos l(s)\\&\quad + \sin (d(p,{\tilde{y}}_2)+s) (\cos l(s))'|\,ds\le C\delta ^\frac{1}{500n^2}. \end{aligned} \end{aligned}$$

(88)

Here, we have

$$\begin{aligned} \begin{aligned}&\cos d(p,\tilde{y_1})-\cos (d(p,{\tilde{y}}_2)+s)\cos l(s)+ \sin (d(p,{\tilde{y}}_2)+s) (\cos l(s))'\\&\quad = \sin (d(p,{\tilde{y}}_2)+s) ( \cos l(s)-\cos d(p,\tilde{y_1})\cos (d(p,{\tilde{y}}_2)+s) )'\\&\quad -(\sin (d(p,{\tilde{y}}_2)+s))' ( \cos l(s)-\cos d(p,\tilde{y_1})\cos (d(p,{\tilde{y}}_2)+s))\\&\quad = (\sin (d(p,{\tilde{y}}_2)+s))^2\left( \frac{\cos l(s)-\cos d(p,\tilde{y_1})\cos (d(p,{\tilde{y}}_2)+s)}{\sin (d(p,{\tilde{y}}_2)+s)}\right) '. \end{aligned} \end{aligned}$$

(89)

Since we have $m_i\pi +\delta ^\frac{1}{2000n^2}-C\delta ^\frac{1}{250n^2} \le d(p,{\tilde{y}}_2)+s\le (m_i+1/2)\pi +C \delta ^\frac{1}{250n^2}$ for all $s\in [0,d({\tilde{y}}_2,{\tilde{y}}_3)]$, we can assume that $m_i\pi +\frac{1}{2}\delta ^\frac{1}{2000n^2}\le d(p,{\tilde{y}}_2)+s\le (m_i+1)\pi -\frac{1}{2}\delta ^\frac{1}{2000n^2}$ by taking $\delta$ sufficient small. Thus, we have

$$\begin{aligned} \frac{1}{(\sin (d(p,{\tilde{y}}_2)+s))^2}\le C \delta ^{-\frac{1}{1000n^2}}. \end{aligned}$$

(90)

By (88), (89), (90) and Claim A.9, we get

$$\begin{aligned} \begin{aligned}&\Big |\cos d(y_1,y_2)-\cos d(p,y_1)\cos d(p,y_2)\\&\quad -(-1)^{m_i}\sin d(p,y_2)\cos d(y_1,\phi _{i j}(y_2)) \Big |\le C\delta ^\frac{1}{1000n^2}. \end{aligned} \end{aligned}$$

(91)

Similarly, we have (91) for the case $d(y_2,x_i)> d(y_2,x_j)$. Using (91) for the pair $(\phi _{i j}(y_2),y_1)$, we get

$$\begin{aligned} \begin{aligned}&\Big |\cos d(\phi _{i j}(y_2),y_1)-\cos d(p,\phi _{i j}(y_2))\cos d(p,y_1) \\&\quad -(-1)^{m_i}\sin d(p,y_1)\cos d(\phi _{i j}(y_2),\phi _{i j}(y_1)) \Big |\le C\delta ^\frac{1}{1000n^2}. \end{aligned} \end{aligned}$$

(92)

By Claim A.9 (i) we have $|\cos d(p,\phi _{i j}(y_2))|\le C\delta ^\frac{1}{250n^2}$. Combining this, (91) and (92), we get the claim. $\square$

The following claim shows that $\psi _{i j}$ satisfies one of the property of the Hausdorff approximation map [Definition 2.3 (i)].

Claim A.11

We have

$$\begin{aligned} |d(y_1,y_2)-d(\psi _{i j}(y_1),\psi _{i j}(y_2)) |\le C(n,K,D)\delta ^\frac{1}{2000n^2} \end{aligned}$$

for all $y_1,y_2\in B_{i j}$.

Proof of Claim A.11

Since $d(y_1,y_2)\le \pi +C\delta ^\frac{1}{250n^2}$ and $d(\phi _{i j}(y_2),\phi _{i j}(y_1))\le \pi +C\delta ^\frac{1}{250n^2}$, we get the claim by Lemma 2.18 and Claim A.10. $\square$

Finally, we show that $\psi _{i j}$ satisfies the other property of the Hausdorff approximation map.

Claim A.12

$\psi _{ij}(B_{i j})$ is $C(n,K,D)\delta ^\frac{1}{4000n^2}$-dense in $S^0*Z_{i j}$ (see Definition 2.3).

Proof of Claim A.12

Take arbitrary $[t,z]\in S^0*Z_{i j}$ with $2\delta ^\frac{1}{2000n^2}<t<\pi -2\delta ^\frac{1}{2000n^2}$. Suppose that $t\ge \pi /2$. Put $x=\gamma _{z,x_j}(t-\pi /2)$. Since we have $d(x,z)=t-\pi /2$, $|d(x_i,z)-\pi /2|\le C\delta ^\frac{1}{250n^2}$ and $|d(x_i,x)-t|\le C\delta ^\frac{1}{250n^2}$, we get $x\in G_{i j}$ by taking $\delta$ sufficient small, and by Claim A.10,

$$\begin{aligned} \left| \cos \left( t-\frac{\pi }{2}\right) -\sin t \cos d(\phi _{i j}(x),z) \right| \le C\delta ^\frac{1}{1000n^2}. \end{aligned}$$

Since $\cos \left( t-\pi /2\right) =\sin t$ and $1/|\sin t|\le C\delta ^{-\frac{1}{2000n^2}}$,

$$\begin{aligned} \left| 1- \cos d(\phi _{i j}(x),z) \right| \le C\delta ^\frac{1}{2000n^2}. \end{aligned}$$

Thus, we get

$$\begin{aligned} |\cos d(\psi _{i j}(x),[t,z])-1|\le C\delta ^\frac{1}{2000n^2}. \end{aligned}$$

By Lemma 2.18, we get $d(\psi _{i j}(x),[t,z])\le C\delta ^\frac{1}{4000n^2}$. Similarly, putting $x=\gamma _{z,x_i}(\pi /2-t)$, we have $d(\psi _{i j}(x),[t,z])\le C\delta ^\frac{1}{4000n^2}$ for the case $t<\pi /2$. Since

$$\begin{aligned} \left\{ [t,z]\in S^0*Z_{i j}: 2\delta ^\frac{1}{2000n^2}<t<\pi -2\delta ^\frac{1}{2000n^2} \text { and } z\in Z_{i j}\right\} \end{aligned}$$

is $2\delta ^\frac{1}{2000n^2}$-dense in $S^0*Z_{i j}$, we get the claim. $\square$

By Claim A.11 and Claim A.12, we get that the map $\psi _{ij}:B_{ij}\rightarrow S^0*Z_{ij}$ is a $C(n,k,D)\delta ^\frac{1}{4000n^2}$-Hausdorff approximation. Thus, we get the proposition. $\square$

Rewording Propositions 2.19 and A.5 by using the limit space, we immediately get the following corollary.

Corollary A.13

Take an integer $n\ge 2$, positive real numbers $K>0$ and $D>0$ and a sequence of positive real numbers $\{\delta _i\}$ $(i\in {\mathbb {N}})$ with $\delta _i\rightarrow 0$. Suppose that a sequence of n-dimensional closed Riemannian manifolds $\{(M_i,g_i)\}$ $(i\in {\mathbb {N}})$ with $\mathop {\mathrm {Ric}}\nolimits _i\ge -Kg_i$ and $\mathop {\mathrm {diam}}\nolimits (M_i)\le D$ converges to a geodesic space (X, d) in the Gromov–Hausdorff topology, and there exists a non-zero function $f_i\in C^\infty (M_i)$ with $\Vert (\nabla ^{g_i})^2 f_i+f_i g_i\Vert _2\le \delta _i \Vert f_i\Vert _2$ for each i. Then, there exist finite points $x_0,\ldots ,x_N\in X$ $(N\le N_3)$ such that the following properties holds.

• (i) For each $i,j=0,\ldots , N$, there exists an integer $k_{ij}\in {\mathbb {Z}}_{\ge 0}$ with $d(x_i,x_j)=\pi k_{ij}$. Moreover, $k_{ij}$ is even if and only if $(-1)^{k_{0i}}=(-1)^{k_{0j}}$ holds.

• (ii) For each $i,j=0,\ldots , N$ with $d(x_i,x_j)=\pi$, we define

  $$\begin{aligned} \begin{aligned} B_{ij}&:=\{x\in X: d(x_i,x)+d(x,x_j)= d(x_i,x_j)\}\\ {\widetilde{B}}_{ij}&:=B_{ij}{\setminus } \{x_i,x_j\}. \end{aligned} \end{aligned}$$

  Then, $X=\bigcup _{ij} B_{ij}$ holds, and ${\widetilde{B}}_{ij}\cap B_{kl}=\emptyset$ if $\{k,l\}\ne \{i,j\}$.

• (iii) Suppose that $i,j,k=0,\ldots , N$ satisfy $j\ne k$, $d(x_i,x_j)=\pi$ and $d(x_i,x_k)=\pi$. Then, for any $x\in B_{ij}$ and $y\in B_{ik}$, we have $d(x,y)=d(x,x_i)+d(x_i,y)$.

• (iv) For each $i,j=0,\ldots , N$ with $d(x_i,x_j)=\pi$, there exists a metric space $Z_{ij}$ such that $B_{ij}$ is isometric to the spherical suspension $S^0*Z_{ij}$.

More detailed consideration enable us to exclude the case $N\ge 2$.

Corollary A.14

Take an integer $n\ge 2$, positive real numbers $K>0$ and $D>0$ and a sequence of positive real numbers $\{\delta _i\}$ $(i\in {\mathbb {N}})$ with $\delta _i\rightarrow 0$. Suppose that a sequence of n-dimensional closed Riemannian manifolds $\{(M_i,g_i)\}$ $(i\in {\mathbb {N}})$ with $\mathop {\mathrm {Ric}}\nolimits _i\ge -Kg_i$ and $\mathop {\mathrm {diam}}\nolimits (M_i)\le D$ converges to a geodesic space (X, d) in the Gromov–Hausdorff topology, and there exists a non-zero function $f_i\in C^\infty (M_i)$ with $\Vert (\nabla ^{g_i})^2 f_i+f_i g_i\Vert _2\le \delta _i \Vert f_i\Vert _2$ for each i. Then, we always have $N=1$ in Corollary A.13, and there exists a compact metric space Z such that $X=S^0*Z$.

Proof

If $N=1$ holds in Corollary A.13, we get the corollary.

In the following, we suppose that $N\ge 2$ holds in Corollary A.13 and show a contradiction.

Claim A.15

Take arbitrary $i=0,\ldots ,N$. Take $j_1,\ldots ,j_l=0,\ldots , N$ such that $j_a\ne j_b$ (if $a\ne b$) and

$$\begin{aligned} \{x_{j_1},\ldots ,x_{j_l}\}=\{x_k:d(x_i,x_k)=\pi \}. \end{aligned}$$

If $l\ge 2$, then we have $l=2$ and $\mathop {\mathrm {Card}}\nolimits Z_{i j_1}= \mathop {\mathrm {Card}}\nolimits Z_{i j_2}=1$.

Proof of Claim A.15

The tangent cone $X_{x_i}$ at $x_i$ is isometric to

$$\begin{aligned} (C(Z_{i j_1}) \coprod \cdots \coprod C(Z_{i j_l}))/\sim , \end{aligned}$$

where $C(Z_{ij_1}),\ldots , C(Z_{i j_l})$ denote the metric cones, and $\sim$ identify the vertexes. Let $0^*$ denotes the vertex. The metric d on $(C(Z_{i j_1}) \coprod \cdots \coprod C(Z_{i j_l}))/\sim$ satisfies $d(c_1,c_2)=d(0^*,c_1)+d(0^*,c_2)$ for all $c_1\in C(Z_{i j_a})$ and $c_2\in C(Z_{i j_b})$ ($a\ne b$).

Suppose that we have either $l\ge 3$ or $\mathop {\mathrm {Card}}\nolimits Z_{i j_a}\ge 2$ for some a. Then, there exist $a\in \{1,\ldots ,l\}$ and points $x_1\in Z_{i j_1}$, $x_2\in Z_{i j_2}$, $x_3\in Z_{i j_a}$ such that $x_3\notin \{x_1,x_2\}$. If necessary, we can exchange $j_1$ and $j_2$ and assume $a\ne 1$. Consider a line $\gamma :{\mathbb {R}}\rightarrow X_{x_i}$ defined by

$$\begin{aligned} \gamma (t)=\left\{ \begin{array}{cc} {[}-t, x_1]\in C(Z_{i j_1}) &{}\quad (t<0),\\ {[}t, x_2]\in C(Z_{i j_2}) &{} \quad (t>0),\\ 0^*&{} \quad (t=0). \end{array}\right. \end{aligned}$$

By the splitting theorem [6, Theorem 9.27], $X_{x_i}$ splits into ${\mathbb {R}}\times Z$ for some geodesic space Z, and there exists a point $y_0\in Z$ such that $\gamma (t)$ corresponds to $(t,y_0)$ for each $t\in {\mathbb {R}}$. In particular, under this isometry, we have

$$\begin{aligned} \left\{ \begin{array}{cc} {[}1,x_1]\in C(Z_{i j_1}) &{}\quad \mapsto (-1,y_0),\\ {[}1, x_2]\in C(Z_{i j_2}) &{}\mapsto (1,y_0),\\ 0^*&{}\quad \mapsto (0,y_0). \end{array}\right. \end{aligned}$$

Take $(s,y)\in {\mathbb {R}}\times Z$ corresponding to $[1,x_3]$. Then, we have

$$\begin{aligned} 1&=d(0^*,[1,x_3])^2=s^2+d(y,y_0)^2,\\ 4&=d([1,x_1],[1,x_3])^2=(s+1)^2+d(y,y_0)^2. \end{aligned}$$

Thus, we get $s=1$ and $d(y,y_0)=0$. This contradicts to $x_2\ne x_3$. Therefore, we get $l=2$ and $\mathop {\mathrm {Card}}\nolimits Z_{i j_1}= \mathop {\mathrm {Card}}\nolimits Z_{i j_2}$=1. $\square$

By the connectedness, there exists i such that $\mathop {\mathrm {Card}}\nolimits \{x_k:d(x_i,x_k)=\pi \}=2$, and we can connect $x_i$ to $x_j$ for any j, i.e., there exists $x_{j_0},\ldots , x_{j_l}$ such that $j_0=i$, $j_l=j$ and $d(x_{j_a},x_{j_{a+1}})=\pi$ for all $a=0,\ldots ,l-1$. Therefore, we get $X=[0,N\pi ]$ or $X=S^1((N+1)/2)$ (c.f. [10, Theorem 1.1] and [27, Theorem 1.1]). Moreover, if $X=S^1((N+1)/2)$, then N is odd by the following argument. Suppose that N is even. Then, $(N+1)/2$ is not a integer, and there exists $x,x_i,x_j\in X$ with $d(x_i,x_j)=\pi$, $d(x_0,x)=(N+1)\pi /2$ and $x\in B_{i j}$. We get $d(x_0,x_i)=d(x_0,x_j)=N\pi /2$, and this contradicts to Corollary A.13 (i). Thus, we have that N is odd.

For each $i\in {\mathbb {N}}$, take $p_i\in M_i$ of Proposition 2.15 for $f_i$. By taking a subsequence, we can assume that there exists a point $p\in X$ such that $p_i{\mathop {\rightarrow }\limits ^{GH}} p$, and X has a limit measure $\nu$. By [27, Theorem 1.1] (see also [27, Definition 2.1]), the limit measure $\nu$ on X is of the form

$$\begin{aligned} \nu =\phi d H^1, \end{aligned}$$

where $H^1$ denotes the 1-dimensional Hausdorff measure and $\phi :X\rightarrow {\mathbb {R}}_{\ge 0}\cup \{\infty \}$ satisfies

$$\begin{aligned} \begin{aligned}&\phi ^{\frac{1}{n}}(\gamma _{x,y}(t)) \ge \frac{1}{\sinh \left( d(x,y)\sqrt{\frac{K}{n}}\right) } \Biggr (\sinh \left( d(\gamma _{x,y}(t),y)\sqrt{\frac{K}{n}}\right) \phi ^{\frac{1}{n}}(x)\\&\quad \qquad +\sinh \left( d(x,\gamma _{x,y}(t))\sqrt{\frac{K}{n}}\right) \phi ^{\frac{1}{n}}(y)\Biggr ) \end{aligned} \end{aligned}$$

(93)

for all $x,y\in X$ and $0\le t\le d(x,y)$. In particular, $\phi$ is a locally Lipschitz function on $(0,N \pi )$ if $X=[0,N\pi ]$, and $\phi$ is a Lipschitz function if $X=S^1(m)$. Set ${\widetilde{f}}_i:=T_{(n,\delta )}(f_i)\in C^\infty (M_i)$ and normalize it so that $\Vert {\widetilde{f}}_i\Vert _2^2=\frac{1}{n+1}$. Taking a subsequence, we can assume that there exists $f\in {\mathcal {D}}(X,\nu )$ such that ${\widetilde{f}}_i$ converges to f strongly in $L^2$, and $\Delta {\widetilde{f}}_i$ converges to $\Delta f$ weakly in $L^2$ by Theorem 1.3 and Theorem 4.9 of [24]. Moreover, $\cos d_i(p_i,\cdot )$ converges to $\cos d(p,\cdot )$ strongly in $L^2$ because we can easily verify the property of [24, Definition 3.7](see also [24, Proposition 3.32]). Thus, we get

$$\begin{aligned} \Vert f-\cos d(p,\cdot )\Vert _2\le \liminf _{i\rightarrow \infty }\Vert {\widetilde{f}}_i-\cos d(p_i,\cdot )\Vert _2=0, \end{aligned}$$

and so $f=\cos d(p,\cdot )$. Since $\Delta {\widetilde{f}}_i-n {\widetilde{f}}_i$ converges to $\Delta f-n f$ weakly in $L^2$, we get

$$\begin{aligned} \Vert \Delta f-n f\Vert _2\le \liminf _{i\rightarrow \infty }\Vert \Delta {\widetilde{f}}_i-n {\widetilde{f}}_i\Vert _2=0, \end{aligned}$$

and so $\Delta f=n f$.

If $X=[0,N\pi ]$, we parameterize $(0,N\pi )\subset [0,N\pi ]$ by the identity map $(0,N\pi )\rightarrow (0,N\pi ),\, \theta \mapsto \theta$. Since $0\in [0,N\pi ]$ corresponds to some $x_i$ of Corollary A.13, we have $f(\theta )=\pm \sin \theta$ under the parameterization. Considering $-f$, if necessary, we can assume that $f(\theta )=\sin \theta$. If $X=S^1(m)$ and $p=m \exp (\sqrt{-1}\theta _0)$, we parameterize $S^1(m){\setminus } \{p\}\subset S^1(m)$ by the map

$$\begin{aligned} (0,2m\pi )\rightarrow S^1(m){\setminus } \{p\},\,\theta \mapsto m \exp (\sqrt{-1}(\theta _0+\theta /m)). \end{aligned}$$

For both cases, we have $f(\theta )=\sin \theta$ under our parameterization.

Take arbitrary $\psi \in C^\infty _0((0,N\pi ))$ if $X=[0,N\pi ]$ and $\psi \in C^\infty _0(S^1(m){\setminus }\{p\})$ if $X=S^1(m)$, where $C^\infty _0$ denotes the space of smooth functions with compact support. Then, we have

$$\begin{aligned} \int _X \langle d f, d\psi \rangle \,d \nu&=-\int \sin \theta \psi ' (\theta ) \phi (\theta )\,d\theta \\&=-\int (\sin \theta \psi (\theta ) \phi (\theta ))'\,d\theta +\int (\sin \theta \phi (\theta ))'\psi (\theta )\,d\theta \\&=\int (\cos \theta \phi (\theta )+ \sin \theta \phi '(\theta ))\psi (\theta )\,d\theta \end{aligned}$$

By the definition of the Laplacian $\Delta$, we have

$$\begin{aligned} \int _X \langle d f, d\psi \rangle \,d \nu =\int _X \Delta f \psi \,d\nu =\int n \cos \theta \phi (\theta )\psi (\theta ) \,d\theta . \end{aligned}$$

Thus, we get

$$\begin{aligned} \sin \theta \phi ' (\theta )=(n-1)\cos \theta \phi (\theta ) \end{aligned}$$

for almost all $\theta$. This gives

$$\begin{aligned} \sin \theta \left( \phi (\theta )-\phi \left( \frac{\pi }{2}\right) \sin ^{n-1}\theta \right) '=(n-1)\cos \theta \left( \phi (\theta )-\phi \left( \frac{\pi }{2}\right) \sin ^{n-1}\theta \right) \end{aligned}$$

for almost all $\theta \in (0,\pi )$. Since $\sin \theta >0$ for any $\theta \in (0,\pi )$, we get $\phi (\theta )=\phi \left( \frac{\pi }{2}\right) \sin ^{n-1}\theta$ on $(0,\pi )$. Similarly, we have $\phi (\theta )=\phi \left( \frac{3\pi }{2}\right) \sin ^{n-1}(\theta -\pi )$ on $(\pi ,2\pi )$. By the Lipschitz continuity of $\phi$, we get $\phi (\pi )=0$. Putting $x=\pi /2$, $y=3\pi /2$ and $t=\pi$ into (93), we get

$$\begin{aligned} \phi (\pi )\ge \frac{1}{\sinh \left( \pi \sqrt{\frac{K}{n}}\right) } \Biggr (\sinh \left( \frac{\pi }{2} \sqrt{\frac{K}{n}}\right) \phi ^{\frac{1}{n}}\left( \frac{\pi }{2}\right) +\sinh \left( \frac{\pi }{2} \sqrt{\frac{K}{n}}\right) \phi ^{\frac{1}{n}}\left( \frac{3\pi }{2}\right) \Biggr )>0. \end{aligned}$$

This is a contradiction. Thus, we get $N=1$. $\square$

By Corollary A.14, we always have $N=1$ in Proposition 2.19 if we take $\delta$ small enough.

Proposition A.16

Given an integer $n\ge 2$ and positive real numbers $K>0$ and $D>0$, there exists a positive constant $\eta (n,K,D)>0$ such that the following property hold. Take a positive real number $0<\delta \le \eta$. Let (M, g) be an n-dimensional closed Riemannian manifold with $\mathop {\mathrm {Ric}}\nolimits \ge -Kg$ and $\mathop {\mathrm {diam}}\nolimits (M)\le D$. Suppose that a non-zero function $f\in C^\infty (M)$ satisfies $\Vert \nabla ^2 f+f g\Vert _2\le \delta \Vert f\Vert _2$. Take a point $p\in M$ of Proposition 2.15. Then, there exists a point $q\in M$ such that $|d(p,q)-\pi |\le \delta ^{\frac{1}{200n^2}}$ and $d(p,x)+d(x,q)\le d(p,q)+\delta ^\frac{1}{250n^2}$ holds for all $x\in M$.

To prove Theorem A.1, let us turn to the consideration of the $\delta$-pinching condition for a subspace $V\subset C^\infty (M)$. For the standard sphere $S^n\subset {\mathbb {R}}^{n+1}$, the height functions are the first eigenfunction, and if $f_1$ and $f_2$ are height functions that are orthogonal to each other in $L^2$ sense, then the distance between the maximum points of these functions is equal to $\pi /2$. The following lemma asserts that such a property almost holds under our $\delta$-pinching condition.

Lemma A.17

Given an integer $n\ge 2$ and positive real numbers $K>0$ and $D>0$, there exists a positive constant $\eta (n,K,D)>0$ such that the following properties hold. Take a positive real number $0<\delta \le \eta$. Let (M, g) be an n-dimensional closed Riemannian manifold with $\mathop {\mathrm {Ric}}\nolimits \ge -Kg$ and $\mathop {\mathrm {diam}}\nolimits (M)\le D$. Suppose that a 2-dimensional subspace $V\subset C^\infty (M)$ satisfies the $\delta$-pinching condition. Take $f_1,f_2\in T_{(n,\delta )}(V)$ such that

$$\begin{aligned} \Vert f_1\Vert _2^2=\Vert f_2\Vert _2^2=\frac{1}{n+1},\quad \int _M f_1 f_2 \,d\mu _g=0. \end{aligned}$$

For each $f_i$ $(i=1,2)$, we use the notation $p_i,q_i$ of Proposition A.16. Then, we have

$$\begin{aligned} \left| d(p_1,p_2)-\frac{\pi }{2}\right| \le C\delta ^\frac{1}{96n},\quad \left| d(p_1,q_2)-\frac{\pi }{2}\right| \le C\delta ^\frac{1}{250n^2}. \end{aligned}$$

Proof

Set $f_0:=f_1+f_2$ and use the notation $p_0$ of Proposition 2.15 for $f_0$. Note that $\Vert f_0\Vert _2^2=\frac{2}{n+1}$.

By Proposition 2.15, we have

$$\begin{aligned} \begin{aligned} |f_0(x)-\sqrt{2}\cos d(p_0,x)|&\le C \delta ^\frac{1}{48n},\\ |f_1(x)-\cos d(p_1,x)|&\le C \delta ^\frac{1}{48n},\\ |f_2(x)-\cos d(p_2,x)|&\le C\delta ^\frac{1}{48n} \end{aligned} \end{aligned}$$

(94)

for all $x\in M$. Combining (94) with $f_0=f_1+f_2$, we get

$$\begin{aligned} \left| \sqrt{2}\cos d(p_0,x)-\cos d(p_1,x)-\cos d(p_2,x)\right| \le C \delta ^\frac{1}{48n} \end{aligned}$$

(95)

for all $x\in M$. Putting $x=p_0, p_1,p_2$ into (95), we get

$$\begin{aligned} \begin{aligned} \left| \sqrt{2}-\cos d(p_0,p_1)-\cos d(p_0,p_2)\right|&\le C \delta ^\frac{1}{48n},\\ \left| \sqrt{2}\cos d(p_0,p_1)-1-\cos d(p_1,p_2)\right|&\le C \delta ^\frac{1}{48n},\\ \left| \sqrt{2}\cos d(p_0,p_2)-1-\cos d(p_1,p_2)\right|&\le C \delta ^\frac{1}{48n}. \end{aligned} \end{aligned}$$

Thus, we get

$$\begin{aligned} \left| \cos d(p_1,p_2)\right|&\le C\delta ^\frac{1}{48n}. \end{aligned}$$

Since we have $d(p_1,p_2)\le \pi +2\delta ^{\frac{1}{250n^2}}$ by Proposition 2.32, we get

$$\begin{aligned} \left| d(p_1,p_2)-\frac{\pi }{2}\right| \le C\delta ^\frac{1}{96n}. \end{aligned}$$

by Lemma 2.18. Thus, we get

$$\begin{aligned} \left| d(p_1,q_2)-\frac{\pi }{2}\right| \le C\delta ^\frac{1}{250n^2} \end{aligned}$$

by Proposition A.16. $\square$

Now, we are in position to prove Theorem A.1.

Proof of Theorem A.1

Let $\{(M_i,g_i)\}_{i\in {\mathbb {N}}}$ be a sequence of n-dimensional closed Riemannian manifolds such that $\mathop {\mathrm {Ric}}\nolimits _{g_i}\ge -K g_i$, $\mathop {\mathrm {diam}}\nolimits (M_i)\le D$, $\lim _{i\rightarrow \infty }\lambda _k({\bar{\Delta }}^E,M_i)= 0$, and $\{(M_i,g_i)\}_{i\in {\mathbb {N}}}$ converges to a geodesic space X.

By Corollary A.14 and Lemma A.17, we have $\mathop {\mathrm {diam}}\nolimits (X)=\pi$, and there exist pairs of points $(p_1,q_1),\ldots ,(p_k,q_k)$ such that the following properties hold:

• (i) $d(p_i,q_i)=\pi$ for all i.

• (ii) $d(p_i,p_j)=d(p_i,q_j)=\frac{\pi }{2}$ if $i\ne j$.

• (iii) Define $Z_i:=\{x\in X:d(x,p_i)=d(x,q_i)\}$. For all i, we define $\phi _i :X\rightarrow Z_i$ as follows. If $x\in X$ satisfies $d(x,p_i)\le d(x,q_i)$, we define $\phi _i(x)$ to be $\phi _i(x)\in \mathop {\mathrm {Im}}\nolimits \gamma _{x,q_i}\cap Z_{i}$. If $d(x,p_i)> d(x,q_i)$, we define $\phi _{i}(x)$ to be $\phi _{i}(x)\in \mathop {\mathrm {Im}}\nolimits \gamma _{x,p_i}\cap Z_{i}$. Then, we have

  $$\begin{aligned} \cos d(x,y)=\cos d(p_i,x)\cos d(p_i,y)+\sin d(p_i,x)\sin d(p_i,y)\cos d(\phi _i(x),\phi _i(y))\nonumber \\ \end{aligned}$$

  (96)

  for all $x,y\in X$.

Note that for all i, we have an isometry $X\cong S^0*Z_i$ such that $p_i\mapsto 0^*$ and $q_i\mapsto \pi ^*$. Under this identification, we have $\phi _i([t,z])=[\pi /2,z]$ for all $[t,z]\in S^0*Z_i$ with $0<t<\pi$.

Claim A.18

Take arbitrary i and $x,y\in Z_i$ with $d(x,y)<\pi$. Then, we have $\gamma _{x,y}(s)\in Z_i$ for all $s\in [0,d(x,y)]$.

Proof of Claim A.18

We have

$$\begin{aligned} \begin{aligned} \cos d(x,\gamma _{x,y}(s))&=\sin d(p_i,\gamma _{x,y}(s))\cos d(x,\phi _i(\gamma _{x,y}(s))),\\ \cos d(y,\gamma _{x,y}(s))&=\sin d(p_i,\gamma _{x,y}(s))\cos d(y,\phi _i(\gamma _{x,y}(s))). \end{aligned} \end{aligned}$$

We first show that $\gamma _{x,y}(t)\in Z_i$ for all $t\in [0,d(x,y)]$ with $d(x,\gamma _{x,y}(t))<\frac{\pi }{2}$ and $d(y,\gamma _{x,y}(t))<\frac{\pi }{2}$. Take arbitrary $t\in [0,d(x,y)]$ with $d(x,\gamma _{x,y}(t))<\frac{\pi }{2}$ and $d(y,\gamma _{x,y}(t))<\frac{\pi }{2}$. Suppose that $\gamma _{x,y}(t)\notin Z_i$. Then, we have

$$\begin{aligned} \begin{aligned} 0<\cos d(x,\gamma _{x,y}(t))<&\cos d(x,\phi _i(\gamma _{x,y}(t))),\\ 0<\cos d(y,\gamma _{x,y}(t))<&\cos d(y,\phi _i(\gamma _{x,y}(t))), \end{aligned} \end{aligned}$$

and so

$$\begin{aligned} \begin{aligned} d(x,\gamma _{x,y}(t))>&d(x,\phi _i(\gamma _{x,y}(t))),\\ d(y,\gamma _{x,y}(t))>&d(y,\phi _i(\gamma _{x,y}(t))). \end{aligned} \end{aligned}$$

Thus, we get $d(x,y)=d(x,\gamma _{x,y}(t))+d(y,\gamma _{x,y}(t))> d(x,\phi _i(\gamma _{x,y}(t)))+ d(y,\phi _i(\gamma _{x,y}(t)))$. This is a contradiction. Therefore, we get $\gamma _{x,y}(t)\in Z_i$.

We next show that $\gamma _{x,y}(t)\in Z_i$ for the general case. Put $t_0:=\frac{1}{2}d(x,y)$. Then $d(x,\gamma _{x,y}(t_0))<\frac{\pi }{2}$ and $d(y,\gamma _{x,y}(t_0))<\frac{\pi }{2}$. Thus, we get $\gamma _{x,y}(t_0)\in Z_i$. For all $t\in [0,t_0]$, we have $d(x,\gamma _{x,y}(t))<\frac{\pi }{2}$ and $d(\gamma _{x,y}(t_0),\gamma _{x,y}(t))<\frac{\pi }{2}$, and so we get $\gamma _{x,y}(t)\in Z_i$. Similarly, we get $\gamma _{x,y}(t)\in Z_i$ for all $t\in [t_0,d(x,y)]$. $\square$

We have shown that $X=S^0*Z_1$. To carry out the iteration process, we investigate the structure of $Z_1\cap \cdots \cap Z_{i-1}$.

Claim A.19

Suppose that $k\ge 2$, $2\le i\le k$ and

$$\begin{aligned} Z_1\cap \cdots \cap Z_{i-1}{\setminus }\{p_{i},q_{i}\}\ne \emptyset . \end{aligned}$$

Then, we have $Z_{1}\cap \cdots \cap Z_{i}\ne \emptyset$ and

$$\begin{aligned} Z_1\cap \cdots \cap Z_{i-1}=S^0*(Z_{1}\cap \cdots \cap Z_{i}). \end{aligned}$$

Proof of Claim A.19

For all $x\in Z_1\cap \cdots \cap Z_{i-1}{\setminus }\{p_{i},q_{i}\}$, we have $d(x,p_{i})<\pi$ and $d(x,q_{i})<\pi$, and so we get $\phi _i(x)\in Z_{1}\cap \cdots \cap Z_{i}$ by Claim A.18. Thus, we have $Z_{1}\cap \cdots \cap Z_{i}\ne \emptyset$. Since we have (96) for all $x,y \in Z_1\cap \cdots \cap Z_{i-1}{\setminus }\{p_{i},q_{i}\}$, the map

$$\begin{aligned} Z_1\cap \cdots \cap Z_{i-1}\rightarrow S^0*(Z_{1}\cap \cdots \cap Z_{i}),\,x\mapsto [d(p_i,x),\phi _i(x)] \end{aligned}$$

gives the isomorphism. Note that the surjectivity of this map also follows from Claim A.18. $\square$

For all $2\le i<k$, we have $p_{k}\in Z_1\cap \cdots \cap Z_{i-1}{\setminus }\{p_i,q_i\}$, and so $Z_1\cap \cdots \cap Z_{i-1}{\setminus }\{p_i,q_i\}\ne \emptyset$. Since $Z_1\cap \cdots \cap Z_{k}\subset Z_1\cap \cdots \cap Z_{k-1}{\setminus }\{p_k,q_k\}$, we get

$$\begin{aligned} Z_1\cap \cdots \cap Z_{k}= \emptyset&\iff Z_1\cap \cdots \cap Z_{k-1}{\setminus }\{p_k,q_k\}=\emptyset \\&\iff Z_1\cap \cdots \cap Z_{k-1}=\{p_k,q_k\} \end{aligned}$$

by Claim A.19. Therefore, we get the following inductively by Claim A.19:

$$\begin{aligned} X=\left\{ \begin{array}{cc} S^{k-1} &{} (Z_1\cap \cdots \cap Z_{k}= \emptyset ),\\ S^{k-1}*(Z_1\cap \cdots \cap Z_{k})&{}(Z_1\cap \cdots \cap Z_{k}\ne \emptyset ). \end{array}\right. \end{aligned}$$

Finally, we investigate the structure of $Z_1\cap \cdots \cap Z_{k}$ when $Z_1\cap \cdots \cap Z_{k}\ne \emptyset$.

Claim A.20

Suppose that $Z_1\cap \cdots \cap Z_{k}\ne \emptyset$. Then, the tangent cone $X_{p_k}$ of X at $p_k$ is isometric to ${\mathbb {R}}^{k-1}\times C(Z_1\cap \cdots \cap Z_{k})$. Moreover, if $Z_1\cap \cdots \cap Z_{k}$ is not connected, then we have $\mathop {\mathrm {Card}}\nolimits (Z_1\cap \cdots \cap Z_{k})=2$.

Proof of Claim A.20

For each $i=1,\ldots , k$, we define a map $\psi _i:X\rightarrow Z_1\cap \cdots \cap Z_{i}$ by $\psi _i:=\phi _i\circ \cdots \circ \phi _1$. Define $\psi _0:=\mathop {\mathrm {Id}}\nolimits _X:X\rightarrow X$. For each $r>0$, we define $\Psi _r :X\rightarrow {\mathbb {R}}^k\times C(Z_1\cap \cdots \cap Z_{k})$ by

$$\begin{aligned} \begin{aligned} \Psi _r(x)&:=\left( \frac{1}{r}\left( d(p_1,\psi _0(x))-\frac{\pi }{2}\right) , \ldots ,\frac{1}{r}\left( d(p_{k-1},\psi _{k-2}(x))-\frac{\pi }{2}\right) ,\right. \\&\quad \left. \left[ \frac{1}{r}d(p_k,\psi _{k-1}(x)), \psi _{k}(x)\right] \right) . \end{aligned} \end{aligned}$$

Note that we have $\Psi _r(p_k)=(0,\ldots ,0,0^*)$. In the following, we show that $(X,p_k,r^{-1}d)$ converges to $({\mathbb {R}}^k\times C(Z_1\cap \cdots \cap Z_{k}),(0,\ldots ,0,0^*))$ in the pointed Gromov–Hausdorff sense through the map $\Psi _r$.

Fix $R>0$. Let us show that

$$\begin{aligned} \Psi _r:\left( B_{rR}(p_k),r^{-1}d \right) \rightarrow B_{R+r}\left( (0,\ldots ,0,0^*)\right) \subset {\mathbb {R}}^k\times C(Z_1\cap \cdots \cap Z_{k}) \end{aligned}$$

is a Hausdorff approximation map for sufficiently small $r>0$. Here, $B_{rR}(p_k)$ denotes the open metric ball for the original metric on (X, d). Note that if this was proved, we can easily modify the map $\Psi _r$ to a Hausdorff approximation map $\left( B_{rR}(p_k),r^{-1} d\right) \rightarrow B_{R}\left( (0,\ldots ,0,0^*)\right)$.

For all $x,y\in B_{r R}(p_k)$, we have

$$\begin{aligned} d(\phi _1(x),\phi _1(y))\le d(x,\phi _1(x))+d(x,y)+d(y,\phi _1(y))\le 4r R. \end{aligned}$$

Since $\psi _i(p_k)=p_k$ for $i=0,\ldots ,k-1$, we have

$$\begin{aligned} d(\psi _i(x),\psi _i(y))\le C(R,k)r \end{aligned}$$

for all $x,y\in B_{r R}(p_k)$ and $i=0,\ldots ,k-1$ inductively. In particular, $\psi _i(x),\psi _i(y)\in B_{C(R,k)r}(p_k)$. For all $a>0$, we use the following notation:

$$\begin{aligned} f=O(r^a):\iff |f|\le C(R,k)r^a. \end{aligned}$$

For all $1\le i \le k-1$ and $x,y\in B_{r R}(p_k)$, we have

$$\begin{aligned} \begin{aligned}&\cos d(\psi _{i-1} (x),\psi _{i-1}(y))\\&\quad =\cos d(p_{i},\psi _{i-1} (x))\cos d(p_i,\psi _{i-1} (y))\\&\qquad +\sin d(p_i,\psi _{i-1} (x)) \sin d(p_i,\psi _{i-1} (y))\cos d(\psi _{i} (x),\psi _{i} (x))\\&\quad =\left( \frac{\pi }{2}-d(p_i, \psi _{i-1}(x))+O(r^3)\right) \left( \frac{\pi }{2}-d(p_i, \psi _{i-1}(y))+O(r^3)\right) \\&\qquad +\left( 1-\frac{1}{2}\left( d(p_i, \psi _{i-1}(x))-\frac{\pi }{2}\right) ^2+O(r^4)\right) \times \\&\quad \,\left( 1-\frac{1}{2}\left( d(p_i, \psi _{i-1}(y))-\frac{\pi }{2}\right) ^2+O(r^4)\right) \left( 1-\frac{1}{2}d(\psi _i(x), \psi _i(y))^2+O(r^4)\right) \\&\quad =1-\frac{1}{2}\big (d(p_i,\psi _{i-1}(x))-d(p_i,\psi _{i-1}(y))\big )^2-\frac{1}{2}d(\psi _i(x), \psi _i(y))^2+O(r^4) \end{aligned} \end{aligned}$$

(97)

by (96), $\frac{\pi }{2}-d(p_i, \psi _{i-1}(x))=O(r)$ and $\frac{\pi }{2}-d(p_i, \psi _{i-1}(y))=O(r)$. By (97) and

$$\begin{aligned} \cos d(\psi _{i-1} (x),\psi _{i-1}(y))=1-\frac{1}{2}d(\psi _{i-1} (x),\psi _{i-1}(y))^2+O(r^4), \end{aligned}$$

we get

$$\begin{aligned} d(\psi _{i-1} (x),\psi _{i-1}(y))^2=\big (d(p_i,\psi _{i-1}(x))-d(p_i,\psi _{i-1}(y))\big )^2+d(\psi _i(x), \psi _i(y))^2+O(r^4). \end{aligned}$$

By induction, we get

$$\begin{aligned} \begin{aligned} d(x,y)^2&=\sum _{i=1}^{k-1}\Big (d(p_i,\psi _{i-1}(x))-d(p_i,\psi _{i-1}(y))\Big )^2\\&\quad +d(\psi _{k-1}(x), \psi _{k-1}(y))^2+O(r^4). \end{aligned} \end{aligned}$$

(98)

We have

$$\begin{aligned} \begin{aligned}&\cos d(\psi _{k-1}(x), \psi _{k-1}(y))\\&\quad = \cos d(p_{k},\psi _{k-1} (x))\cos d(p_k,\psi _{k-1} (y))\\&\qquad +\sin d(p_k,\psi _{k-1} (x)) \sin d(p_k,\psi _{k-1} (y))\cos d(\psi _{k} (x),\psi _{k} (y))\\&\quad =\left( 1-\frac{1}{2}d(p_k, \psi _{k-1}(x))^2+O(r^4)\right) \left( 1-\frac{1}{2}d(p_k, \psi _{k-1}(y))^2+O(r^4)\right) \\&\qquad +\left( d(p_k, \psi _{k-1}(x))+O(r^3)\right) \left( d(p_k, \psi _{k-1}(y))+O(r^3)\right) \cos d(\psi _{k}(x), \psi _{k}(y))\\&\quad = 1-\frac{1}{2}d(p_k, \psi _{k-1}(x))^2 -\frac{1}{2}d(p_k, \psi _{k-1}(y))^2\\&\qquad +d(p_k, \psi _{k-1}(x)) d(p_k, \psi _{k-1}(y))\cos d(\psi _{k}(x), \psi _{k}(y))+O(r^4). \end{aligned} \end{aligned}$$

(99)

By (99) and

$$\begin{aligned} \cos d(\psi _{k-1} (x),\psi _{k-1}(y))=1-\frac{1}{2}d(\psi _{k-1} (x),\psi _{k-1}(y))^2+O(r^4), \end{aligned}$$

we get

$$\begin{aligned} \begin{aligned}&d(\psi _{k-1} (x),\psi _{k-1}(y))^2\\&\quad =d(p_k, \psi _{k-1}(x))^2+d(p_k, \psi _{k-1}(y))^2\\&\quad -2d(p_k, \psi _{k-1}(x)) d(p_k, \psi _{k-1}(y))\cos d(\psi _{k}(x), \psi _{k}(y))+O(r^4). \end{aligned} \end{aligned}$$

(100)

By (98) and (100), we get

$$\begin{aligned} \left( \frac{1}{r}d(x,y)\right) ^2 =d(\Psi _r(x),\Psi _r(y))^2+O(r^2). \end{aligned}$$

(101)

In particular, we have $\Psi _r(x),\Psi _r(y)\in B_{R+r}\left( (0,\ldots ,0,0^*)\right) \subset {\mathbb {R}}^k\times C(Z_1\cap \cdots \cap Z_{k})$ for sufficient small $r>0$.

For all $(t_1,\ldots ,t_{k-1},[t_k,z])\in {\mathbb {R}}^{k-1}\times C(Z_1\cap \cdots \cap Z_{k})$, we have

$$\begin{aligned} \Psi _r\left( \left[ r t_1+\frac{\pi }{2},\ldots ,rt_{k-1}+\frac{\pi }{2},r t_k,z\right] \right) =(t_1,\ldots ,t_{k-1},[t_k,z]). \end{aligned}$$

If

$$\begin{aligned} (t_1,\ldots ,t_{k-1},[t_k,z])\in B_{R-r}\left( (0,\ldots ,0,0^*)\right) \subset {\mathbb {R}}^{k-1}\times C(Z_1\cap \cdots \cap Z_{k}), \end{aligned}$$

then $\left[ r t_1+\frac{\pi }{2},\ldots ,rt_{k-1}+\frac{\pi }{2},r t_k,z\right] \in B_{C(R,k)r}(p_k),$ and so

$$\begin{aligned} \left[ r t_1+\frac{\pi }{2},\ldots ,rt_{k-1}+\frac{\pi }{2},r t_k,z\right] \in B_{rR}(p_k) \end{aligned}$$

by (101) if $r>0$ is small enough. This shows that the map $\Psi _r:\left( B_{rR}(p_k),r^{-1}d \right) \rightarrow B_{R+r}\left( (0,\ldots ,0,0^*)\right)$ is 2r-dense. Combining this and (101), we have that $\Psi _r$ is a 2r-Hausdorff approximation map, and so we get the first assertion.

We next show the second assertion. The proof is similar to Corollary A.14. Suppose that there exist more than two different connected components of $Z_1\cap \cdots \cap Z_{k}$ (let A be one of the connected components), and there exist two points $b,c\in Z_1\cap \cdots \cap Z_{k}{\setminus } A$ ($b\ne c$). Take arbitrary $a\in A$. Then, $\gamma :(-\infty ,\infty )\rightarrow C(Z_1\cap \cdots \cap Z_{k})$ defined by $\gamma (t)=[-t,a]$ for $t\le 0$, and $\gamma (t)=[t,b]$ for $t>0$, is a line in $C(Z_1\cap \cdots \cap Z_{k})$. Thus, $(0,\gamma )$ in $X_{p_k}={\mathbb {R}}^{k-1}\times C(Z_1\cap \cdots \cap Z_{k})$ is a line. By the splitting theorem [6, Theorem 9.27], there exists a geodesic space Y, a point $y_0\in Y$ and an isometry $X_{p_k}\rightarrow {\mathbb {R}}\times Y$ such that $(0,\gamma (t))\in X_{p_k}$ corresponds to $(t,y_0)\in {\mathbb {R}}\times Y$ for all $t\in {\mathbb {R}}$. Take a point $(s,y)\in {\mathbb {R}}\times Y$ that corresponds to $(0,[1,c])\in X_{p_k}$. Since $d((0,0^*), (0,[1,c]))=1$, we have $s^2+d(y,y_0)^2=1$. Since $d((0,[1,a]), (0,[1,c]))=2$, and (0, [1, a]) corresponds to $(-1,y_0)$, we have $(s+1)^2+d(y,y_0)^2=4$. Thus, we get that $s=1$ and $d(y,y_0)=0$. However, (0, [1, b]) corresponds to and $(1,y_0)$. Since we assumed that $c\ne b$, this is a contradiction. Thus, we get the second assertion. $\square$

If $\mathop {\mathrm {Card}}\nolimits (Z_1\cap \cdots \cap Z_{k})=2$, then $\mathop {\mathrm {diam}}\nolimits (Z_1\cap \cdots \cap Z_{k})=\pi$ by Claim A.18, and so we get $X=S^{k-1}*\{0,\pi \}=S^k$.

If $Z_1\cap \cdots \cap Z_{k}$ is connected, we define a metric $d_L$ on $Z_1\cap \cdots \cap Z_{k}$ by

$$\begin{aligned} \begin{aligned}&d_L(x,y):=\inf \Big \{\sum _{j=1}d(x_{j-1},x_j): N\in {\mathbb {Z}}_{>0}, \, x_j\in Z_1\cap \cdots \cap Z_{k} \text { for all }j=0,\ldots ,N,\\&\quad x_0=x,\, x_N=y \hbox { and } d(x_{j-1},x_j)<\pi \text { for all}\ j=1,\ldots ,N\Big \}. \end{aligned} \end{aligned}$$

Then, $(Z_1\cap \cdots \cap Z_{k},d_L)$ is a geodesic space, and $X=S^{k-1}*(Z_1\cap \cdots \cap Z_{k},d_L)$ holds.

By the Gromov’s pre-compactness theorem and the above argument, we get the theorem. $\square$

Another Proof of Main Theorem 1

Let $\{(M_i,g_i)\}_{i\in {\mathbb {N}}}$ be a sequence of n-dimensional closed Riemannian manifolds such that $\mathop {\mathrm {Ric}}\nolimits _{g_i}\ge -K g_i$, $\mathop {\mathrm {diam}}\nolimits (M_i)\le D$ for all i and $\lim _{i\rightarrow \infty }\lambda _n({\bar{\Delta }}^E,M_i)= 0$. For each i, define an orientable n-dimensional closed Riemannian manifold $(N_i,{\tilde{g}}_i)$ to be the orientable Riemannian covering of $(M_{i},g_{i})$ with two sheets if $M_i$ is not orientable, and $(N_i,{\tilde{g}}_i)=(M_i,g_i)$ if $M_i$ is orientable. Then, we have $\lim _{i\rightarrow \infty }\lambda _{n+1}({\bar{\Delta }}^E,N_i)= 0$ by Corollary 4.9. Take a subsequence i(j) such that $(M_{i(j)},g_{i(j)})$, $(N_{i(j)},{\tilde{g}}_{i(j)})$ converges to some geodesic spaces X, Y in Gromov–Hausdorff topology, respectively. Since $\dim Y\le n$ holds, we have $Y=S^n$ by Theorem A.1 and [22, Proposition 5.6], where $\dim$ denotes the Hausdorff dimension. In particular, we get $\{(M_{i(j)},g_{i(j)})\}$ is a non-collapsing sequence. Thus, we get $X=S^n$ or $X=S^{n-1}*Z$ for some geodesic space Z by Theorem A.1. By [22, Proposition 5.6], we get that $\mathop {\mathrm {Card}}\nolimits Z=1$ if $X=S^{n-1}*Z$. However, $S^{n-1}*\{\text {point}\}=S^n_{+}$ (n-dimensional hemisphere), and this contradict to [8, Theorem 6.2]. Thus, we get $X=S^n$ and Main Theorem 1. $\square$

Appendix B. Continuity of the eigenvalues

In this appendix we prove the continuity of the eigenvalues of the Laplacian defined in Definition 4.1 for a non-collapsed Gromov–Hausdorff convergent sequence of n-dimensional closed Riemannian manifolds with a uniform 2-sided bound on the Ricci curvature. As an application, on such a limit space, we consider the Obata equation $\nabla ^2 f+fg=0$ and generalize our main theorem.

Take an integer $n\ge 2$, real numbers $K_1,K_2\in {\mathbb {R}}$ with $K_1<K_2$ and positive real numbers $D>0$ and $v>0$. Let ${\mathcal {M}}={\mathcal {M}}(n,K_1,K_2,D,v)$ be the set of isometry classes of n-dimensional closed Riemannian manifolds (M, g) with $K_1g\le \mathop {\mathrm {Ric}}\nolimits _g \le K_2 g$, $\mathop {\mathrm {diam}}\nolimits (M)\le D$ and $\mathop {\mathrm {Vol}}\nolimits (M)\ge v$. Let $\overline{{\mathcal {M}}}=\overline{{\mathcal {M}}}(n,K_1,K_2,D,v)$ be the closure of ${\mathcal {M}}$ in the Gromov–Hausdorff topology. If $X_i\in \overline{{\mathcal {M}}}$ ($i\in {\mathbb {N}}$) converges to $X\in \overline{{\mathcal {M}}}$ in the Gromov–Hausdorff topology, then there exist a sequence of positive real numbers $\{\epsilon _i\}_{i\in {\mathbb {N}}}$ with $\lim _{i\rightarrow \infty }\epsilon _i=0$, and a sequence of $\epsilon _i$-Hausdorff approximation maps $\phi _i :X_i\rightarrow X$. Fix such a sequence. We say a sequence $x_i\in X_i$ converges to $x\in X$ if $\lim _{i\rightarrow \infty }\phi _i(x_i)=x$ (denote it by $x_i{\mathop {\rightarrow }\limits ^{GH}} x$). By the volume convergence theorem [8, Theorem 5.9], $(X_i,H^n)$ converges to $(X,H^n)$ in the measured Gromov–Hausdorff sense, i.e., for all $r>0$ and all sequence $x_i\in X_i$ that converges to $x\in X$, we have $\lim _{i\rightarrow \infty }H^n(B_r (x_i))=H^n(B_r(x))$, where $H^n$ denotes the n-dimensional Hausdorff measure. In particular, for all $X\in \overline{{\mathcal {M}}}$, we have $v\le H^n(X)\le C(n,K_1,D)$.

For all $X\in \overline{{\mathcal {M}}}$, we can consider the cotangent bundle $\pi :T^*X \rightarrow X$ with a canonical inner product by [5, 9] (see also [24, Section 2] for a short review). We have $H^n(X{\setminus } \pi (T^*X))=0$ and $T^*_x X:=\pi ^{-1}(x)$ is an n-dimensional vector space for all $x\in \pi (T^*X)$. For all Lipschitz function f on X, we can define $d f(x)\in T_x^*X$ for almost all $x\in X$, and we have $d f\in L^\infty (T^*X)$. Let TX be the dual bundle of $T^*X$. Let $\langle \cdot ,\cdot \rangle$ denotes the inner product on $T^*X$, TX and tensor bundles. We also denote the inner product on TX by $g=g_X\in L^\infty (T^*X\otimes T^*X)$. We consider the vector bundle $E(X):=T^*X\oplus {\mathbb {R}}e$ on X with the product metric $\langle \cdot ,\cdot \rangle =\langle \cdot ,\cdot \rangle _E$. We have an identification $L^2(E(X))=L^2(T^*X)\oplus L^2(X)$ by considering the map $\omega +f e\mapsto (\omega ,f)$.

Honda [24] (see also page 1595–1596 of [25]) defined the concepts of $L^p$ weakly convergence and $L^p$ strong convergence for functions and tensors fields ($p\in (1,\infty )$). Most of important properties are summarized in page 1596–1598 of [25]. Similarly, we give the following definition.

Definition B.1

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be a sequence in ${\mathcal {M}}$ and let $X\in \overline{{\mathcal {M}}}$ be the Gromov–Hausdorff limit. Take a sequence $T_i\in L^2(E(X_i))$ and $T\in L^2(E(X))$.

• (i) We say that $T_i$ converges to T weakly in $L^2$ if

  $$\begin{aligned} \sup _{i\in {\mathbb {N}}}\Vert T_i\Vert _2<\infty , \end{aligned}$$

  and for all $r>0$, $y_i,z_i\in X_i$ and $y,z\in X$ with $y_i{\mathop {\rightarrow }\limits ^{GH}} y$ and $z_i{\mathop {\rightarrow }\limits ^{GH}}z$, we have

  $$\begin{aligned} \begin{aligned} \lim _{i\rightarrow \infty }\int _{B_r(y_i)} \langle T_i, d r_{z_i}\rangle _E \,d H^n&=\int _{B_r(y)} \langle T, d r_{z}\rangle _E \,d H^n,\\ \lim _{i\rightarrow \infty }\int _{B_r(y_i)} \langle T_i, e\rangle _E \,d H^n&=\int _{B_r(y)} \langle T, e\rangle _E \,d H^n, \end{aligned} \end{aligned}$$

  where $r_z(x):=d(z,x)$ for all $x\in X$.

• (ii) We say that $T_i$ converges to T strongly in $L^2$ if $T_i$ converges to T weakly in $L^2$, and

  $$\begin{aligned} \limsup _{i\rightarrow \infty } \Vert T_i\Vert _2\le \Vert T\Vert _2 \end{aligned}$$

  holds.

We have that $T_i=\omega _i+f_i e\in L^2(E(X_i))$ converges to $T=\omega +f e\in L^2(E(X))$ weakly in $L^2$ if and only if $\omega _i$, $f_i$ converges to $\omega$, f weakly in $L^2$, respectively. Similarly, $T_i\in L^2(E(X_i))$ converges to $T\in L^2(E(X))$ strongly in $L^2$ if and only if $\omega _i$, $f_i$ converges to $\omega$, f strongly in $L^2$, respectively. One of the important property for the weakly convergent sequence is the lower semi-continuity of the norm:

$$\begin{aligned} \liminf _{i\rightarrow \infty }\Vert T_i\Vert _2\ge \Vert T\Vert _2 \end{aligned}$$

holds if $T_i\in L^2(E(X_i))$ converges to $T\in L^2(E(X))$ weakly in $L^2$ (see [24, Proposition 3.64]). Thus, $T_i$ converges to T strongly in $L^2$ if and only if $T_i$ converges to T weakly in $L^2$ and $\lim _{i\rightarrow \infty } \Vert T_i\Vert _2=\Vert T\Vert _2$ holds. For the linearity of the strong convergence, see [24, Corollary 3.59].

We next consider the space $L^2(T^*X \oplus E(X))$. We also have an identification $L^2(T^*X \oplus E(X))=L^2(T^*X\otimes T^*X)\oplus L^2(TX)$, and define the concepts of convergence for $L^2(T^*X\oplus E(X))$.

Definition B.2

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be a sequence in ${\mathcal {M}}$ and let $X\in \overline{{\mathcal {M}}}$ be the Gromov–Hausdorff limit. Take a sequence $T_i\in L^2(T^*X_i\oplus E(X_i))$ and $T\in L^2(T^*X\oplus E(X))$.

• (i) We say that $T_i$ converges to T weakly in $L^2$ if

  $$\begin{aligned} \sup _{i\in {\mathbb {N}}}\Vert T_i\Vert _2<\infty , \end{aligned}$$

  and for all $r>0$, $y_i,z_i^1,z_i^2\in X_i$ and $y,z^1,z^2\in X$ with $y_i{\mathop {\rightarrow }\limits ^{GH}} y$, $z_i^j{\mathop {\rightarrow }\limits ^{GH}}z$ ($j=1,2$), we have

  $$\begin{aligned} \begin{aligned} \lim _{i\rightarrow \infty }\int _{B_r(y_i)} \langle T_i, d r_{z_i^1}\otimes d r_{z_i^2}\rangle _E \,d H^n&=\int _{B_r(y)} \langle T, d r_{z^1}\otimes d r_{z^2}\rangle _E \,d H^n,\\ \lim _{i\rightarrow \infty }\int _{B_r(y_i)} \langle T_i, d r_{z_i^1}\otimes e\rangle _E \,d H^n&=\int _{B_r(y)} \langle T, dr_{z^1}\otimes e\rangle _E \,d H^n. \end{aligned} \end{aligned}$$

• (ii) We say that $T_i$ converges to T strongly in $L^2$ if $T_i$ converges to T weakly in $L^2$ and

  $$\begin{aligned} \limsup _{i\rightarrow \infty } \Vert T_i\Vert _2\le \Vert T\Vert _2 \end{aligned}$$

  holds.

We have that $T_i=S_i+\omega _i\otimes e\in L^2(T^*X_i\otimes E(X_i))$ converges to $T=S+\omega \otimes e\in L^2(T^*X\otimes E(X))$ weakly in $L^2$ if and only if $S_i$, $\omega _i$ converges to S, $\omega$ weakly in $L^2$, respectively. Similarly, $T_i\in L^2(T^*X_i\otimes E(X_i))$ converges to $T\in L^2(T^*X\otimes E(X))$ strongly in $L^2$ if and only if $S_i$, $\omega _i$ converges to S, $\omega$ strongly in $L^2$, respectively.

We list the definitions of important functional spaces and the eigenvalues of Laplacian on limit spaces. Some of them are first introduced by Gigli [19].

Definition B.3

Let $X\in \overline{{\mathcal {M}}}$.

• (i) Let $\mathop {\mathrm {LIP}}\nolimits (X)$ be the set of the Lipschitz functions on X. For all $f\in \mathop {\mathrm {LIP}}\nolimits (X)$, we define $\Vert f\Vert _{H^{1,2}}^2=\Vert f\Vert _2^2+\Vert d f\Vert _2^2$. Let $H^{1,2}(X)$ be the completion of $\mathop {\mathrm {LIP}}\nolimits (X)$ with respect to this norm.

• (ii) Define

  $$\begin{aligned} \begin{aligned} {\mathcal {D}}^2(\Delta ,X):=\Big \{f\in H^{1,2}(X)&: \text { there exists }F\in L^2(X) \text { such that}\\&\int _X \langle df, dh \rangle \,d H^n=\int _X F h\,d H^n \hbox { for all }\ h\in H^{1,2}(X) \Big \}. \end{aligned} \end{aligned}$$

  For any $f\in {\mathcal {D}}^2(\Delta ,X)$, the function $F\in L^2(X)$ is uniquely determined. Thus, we define $\Delta f:=F$.

• (iii) Define

  $$\begin{aligned} \begin{aligned} \mathop {\mathrm {Test}}\nolimits F(X)&:=\left\{ f\in {\mathcal {D}}^2(\Delta ,X)\cap \mathop {\mathrm {LIP}}\nolimits (X):\Delta f\in H^{1,2}(X)\right\} ,\\ \mathop {\mathrm {Test}}\nolimits T^*X&:=\left\{ \sum _{i=1}^N f_i d h_i: N\in {\mathbb {N}},\, f_i, h_i\in \mathop {\mathrm {Test}}\nolimits F(X)\right\} ,\\ \mathop {\mathrm {Test}}\nolimits E(X)&:=\left\{ \sum _{i=1}^N \psi _i d\phi _i+fe: N\in {\mathbb {N}},\, \psi _i, \phi _i,f\in \mathop {\mathrm {Test}}\nolimits F(X)\right\} . \end{aligned} \end{aligned}$$

  We have an identification $\mathop {\mathrm {Test}}\nolimits E(X)=\mathop {\mathrm {Test}}\nolimits T^*X\oplus \mathop {\mathrm {Test}}\nolimits F(X)$.

• (vi) The operator

  $$\begin{aligned} \nabla :\mathop {\mathrm {Test}}\nolimits T^*X\rightarrow L^2(T^*X \otimes T^*X) \end{aligned}$$

  is defined by $\nabla \sum _{i=1}^N f_i d h_i:=\sum _{i=1}^N \left( d f_i\otimes d h_i+ f_i \nabla ^2 h_i \right)$, where $\nabla ^2$ denotes the Hessian $\mathop {\mathrm {Hess}}\nolimits ^g$ defined in [23]. Note that $\nabla \omega \in L^2(T^*X \otimes T^*X)$ for all $\omega \in \mathop {\mathrm {Test}}\nolimits T^*X$ by [26, Theorem 4.11]. We define

  $$\begin{aligned} \nabla ^E:\mathop {\mathrm {Test}}\nolimits E(X)\rightarrow L^2(T^*X\otimes E(X)) \end{aligned}$$

  by $\nabla ^E (\omega +f e):=\nabla \omega +f g+(d f-\omega )\otimes e$.

• (v) We define the norms

  $$\begin{aligned} \begin{aligned} \Vert \omega \Vert _{H_C^{1,2}}^2&:=\Vert \omega \Vert _2^2+\Vert \nabla \omega \Vert _2^2\quad (\omega \in \mathop {\mathrm {Test}}\nolimits T^*X),\\ \Vert T\Vert _{H_E^{1,2}}^2&:=\Vert T\Vert _2^2+\Vert \nabla ^E T\Vert _2^2\quad (T\in \mathop {\mathrm {Test}}\nolimits E(X)). \end{aligned} \end{aligned}$$

  Let $H^{1,2}_C(T^*X)$ and $H^{1,2}_E(E(X))$ be the completion of $\mathop {\mathrm {Test}}\nolimits T^*X$ and $\mathop {\mathrm {Test}}\nolimits E(X)$ with respect to the norms $\Vert \cdot \Vert _{H^{1,2}_C}$ and $\Vert \cdot \Vert _{H^{1,2}_E}$.

• (vi) Define

  $$\begin{aligned} \begin{aligned}&{\mathcal {D}}^2(\Delta _{C,1},X):=\Big \{\omega \in H^{1,2}_C(T^*X) : \text { there exists }{\hat{\omega }}\in L^2(T^*X) \text { such that}\\&\quad \qquad \int _X \langle \nabla \omega , \nabla \eta \rangle \,d H^n=\int _X \langle {\hat{\omega }}, \eta \rangle \,d H^n \hbox { for all }\ \eta \in H_C^{1,2}(T^*X) \Big \}. \end{aligned} \end{aligned}$$

  For any $\omega \in {\mathcal {D}}^2(\Delta _{C,1},X)$, the form ${\hat{\omega }}\in L^2(T^*X)$ is uniquely determined. Thus, we put $\Delta _{C,1} \omega :={\hat{\omega }}$.

• (vii) Define

  $$\begin{aligned} \begin{aligned}&{\mathcal {D}}^2({\bar{\Delta }}^E,X):=\Big \{T\in H^{1,2}_E(E(X)) : \text {there exists} {\hat{T}}\in L^2(E(X)) \text { such that}\\&\quad \qquad \int _X \langle \nabla ^E T, \nabla ^E S \rangle \,d H^n=\int _X \langle {\hat{T}}, S\rangle \,d H^n \hbox { for all }\ S\in H_E^{1,2}(E(X)) \Big \}. \end{aligned} \end{aligned}$$

  For any $T\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$, the section ${\hat{T}}\in L^2(E(X))$ is uniquely determined. Thus, we define ${\bar{\Delta }}^E T:={\hat{T}}$.

• (viii) For all $k\in {\mathbb {Z}}_{>0}$, we define

  $$\begin{aligned} \begin{aligned} \lambda _k({\bar{\Delta }}^{E}):=\inf \left\{ \sup _{T\in {\mathcal {E}}_k{\setminus } \{0\}}\frac{\Vert \nabla ^E T\Vert ^2_2}{\Vert T\Vert ^2_2}: {\mathcal {E}}_k\subset H^{1,2}_E(E(X))\text { is a }k\text {-dimensional subspace}\right\} . \end{aligned} \end{aligned}$$

We give some easy lemmas about Definition B.3.

We first investigate the relationship between $H^{1,2}_C(T^*X)$, $H^{1,2}(X)$ and $H^{1,2}(E(X))$.

Lemma B.4

Let $X\in \overline{{\mathcal {M}}}$. For all $T=\omega + f e\in \mathop {\mathrm {Test}}\nolimits E(X)$, we have

$$\begin{aligned} \frac{1}{2(n+1)}(\Vert \omega \Vert _{H^{1,2}_C}^2+\Vert f\Vert _{H^{1,2}}^2)\le \Vert T\Vert _{H^{1,2}_E}^2\le 2(n+1)(\Vert \omega \Vert _{H^{1,2}_C}^2+\Vert f\Vert _{H^{1,2}}^2), \end{aligned}$$

and so we have an identification $H^{1,2}_E(E(X))=H^{1,2}_C(T^*X)\oplus H^{1,2}(X)$.

Proof

Take arbitrary $T=\omega + f e\in \mathop {\mathrm {Test}}\nolimits E(X)$. We have

$$\begin{aligned} \begin{aligned} \Vert T\Vert _{H^{1,2}_E}&=\Vert T\Vert _2^2+\Vert \nabla ^E T\Vert _2^2\\&\le \Vert \omega \Vert _2^2+\Vert f\Vert _2^2+(\Vert \nabla \omega \Vert _2+\sqrt{n}\Vert f\Vert _2)^2+(\Vert d f\Vert _2+\Vert \omega \Vert _2)^2\\&\le 2(n+1)(\Vert \omega \Vert _{H^{1,2}_C}^2+\Vert f\Vert _{H^{1,2}}^2). \end{aligned} \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \begin{aligned}&(n+1)\Vert T\Vert _{H^{1,2}_E}\\&\quad \ge \Vert \omega \Vert _2^2+\Vert f\Vert _2^2+(\sqrt{n}\Vert f\Vert _2)^2+(\Vert \nabla \omega \Vert _2-\sqrt{n}\Vert f\Vert _2)^2 +\Vert \omega \Vert _2^2+(\Vert d f\Vert _2-\Vert \omega \Vert _2)^2\\&\quad \ge \Vert \omega \Vert _2^2+\Vert f\Vert _2^2+\frac{1}{2}\Vert \nabla \omega \Vert _2^2+\frac{1}{2}\Vert d f\Vert _2^2\\&\quad \ge \frac{1}{2}(\Vert \omega \Vert _{H^{1,2}_C}^2+\Vert f\Vert _{H^{1,2}}^2). \end{aligned} \end{aligned}$$

Thus, we get the lemma. $\square$

We next investigate the relationship between ${\mathcal {D}}^2(\Delta _{C,1},X)$, ${\mathcal {D}}^2(\Delta ,X)$ and ${\mathcal {D}}^2({\bar{\Delta }}^E,X)$.

Lemma B.5

Let $X\in \overline{{\mathcal {M}}}$. We have an identification

$$\begin{aligned} {\mathcal {D}}^2({\bar{\Delta }}^E,X)={\mathcal {D}}^2(\Delta _{C,1},X)\oplus {\mathcal {D}}^2(\Delta ,X). \end{aligned}$$

Moreover, for all $T=\omega +f e\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$, we have ${\bar{\Delta }}^E T=\Delta _{C,1} \omega -2d f +\omega +(\Delta f -2\delta \omega +n f)e$.

Proof

For all $T=\omega +f e,S=\eta +he\in H^{1,2}_E(E(X))$, we have

$$\begin{aligned} \begin{aligned}&\int _X \langle \nabla ^E T, \nabla ^E S \rangle \,d H^n\\&=\int _{X}\Big ( \langle \nabla \omega ,\nabla \eta \rangle -f\nabla ^*\eta - h\nabla ^*\omega +n f h +\langle d f-\omega ,d h-\eta \rangle \Big ) \,d H^n, \end{aligned} \end{aligned}$$

(102)

where we defined $\nabla ^*\omega :=-\mathop {\mathrm {tr}}\nolimits \nabla \omega \in L^2(X)$. Since $\nabla ^*\omega =\delta \omega$ holds for the smooth case, we have $\nabla ^*\omega =\delta \omega$ for the general case by [24, Theorem 3.74] and the smooth approximation theorem [25, Theorem 3.5], where $\delta \omega \in L^2(X)$ is characterized by satisfying

$$\begin{aligned} \int _X f \delta \omega \,dH^n=\int _X \langle \omega , d f\rangle \,dH^n, \end{aligned}$$

for all $f\in \mathop {\mathrm {Test}}\nolimits F(X)$.

We first prove ${\mathcal {D}}^2(\Delta _{C,1},X)\oplus {\mathcal {D}}^2(\Delta ,X)\subset {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ and the second assertion. Take arbitrary $T=\omega +f e\in {\mathcal {D}}^2(\Delta _{C,1},X)\oplus {\mathcal {D}}^2(\Delta ,X)$. For all $S=\eta +he\in H^{1,2}_E(E(X))$, by (102) we have

$$\begin{aligned} \begin{aligned}&\int _X \langle \nabla ^E T, \nabla ^E S \rangle \,d H^n\\&=\int _{X}\Big ( \langle \Delta _{C,1} \omega ,\eta \rangle -\langle d f, \eta \rangle - h\delta \omega +n f h +\langle \Delta f-\delta \omega , h\rangle -\langle d f-\omega ,\eta \rangle \Big ) \,d H^n\\&=\int _{X} \langle \Delta _{C,1} \omega -2d f +\omega +(\Delta f -2\delta \omega +n f)e,S\rangle \,d H^n. \end{aligned} \end{aligned}$$

Thus, we get $T\in {\mathcal {D}}^2(\Delta ,X)$ and ${\bar{\Delta }}^E T=\Delta _{C,1} \omega -2d f +\omega +(\Delta f -2\delta \omega +n f)e$.

We next prove ${\mathcal {D}}^2({\bar{\Delta }}^E,X)\subset {\mathcal {D}}^2(\Delta _{C,1},X)\oplus {\mathcal {D}}^2(\Delta ,X)$. Take arbitrary $T=\omega +f e\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$. Take ${\hat{\omega }}\in L^2(T^*X)$ and ${\hat{f}}\in L^2(X)$ with ${\bar{\Delta }}^E T={\hat{\omega }}+{\hat{f}} e$. For all $\eta \in H^{1,2}_C(T^*X)$, by (102) we have

$$\begin{aligned} \begin{aligned} \int _X \langle {\hat{\omega }},\eta \rangle \,d H^n&=\int _X \langle \nabla ^E T, \nabla ^E \eta \rangle \,d H^n\\&=\int _{X}\Big ( \langle \nabla \omega ,\nabla \eta \rangle -\langle d f, \eta \rangle +\langle d f-\omega ,-\eta \rangle \Big ) \,d H^n, \end{aligned} \end{aligned}$$

and so

$$\begin{aligned} \int _{X} \langle \nabla \omega ,\nabla \eta \rangle \,d H^n =\int _X \langle {\hat{\omega }}+2d f -\omega ,\eta \rangle \,d H^n. \end{aligned}$$

Thus, we have $\omega \in {\mathcal {D}}^2(\Delta _{C,1},X)$. Similarly, we have $f\in {\mathcal {D}}^2(\Delta ,X)$. $\square$

We mention the convergence of the image of the connection $\nabla ^E$.

Lemma B.6

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be a sequence in ${\mathcal {M}}$ and let $X\in \overline{{\mathcal {M}}}$ be the Gromov–Hausdorff limit. Suppose that a sequence $T_i=\omega _i+f_i e\in H^{1,2}_E(E(X_i))$ converges to $T=\omega +f e\in H^{1,2}(E(X))$ strongly in $L^2$. Then, we have that $\nabla ^E T_i$ converges to $\nabla ^E T$ weakly in $L^2$ if and only if $\nabla \omega _i,d f_i$ converges to $\nabla \omega ,d f$ weakly in $L^2$, respectively. We also have that $\nabla ^E T_i$ converges to $\nabla ^E T$ strongly in $L^2$ if and only if $\nabla \omega _i,d f_i$ converges to $\nabla \omega ,d f$ strongly in $L^2$, respectively.

Proof

Since $\nabla ^E T_i= \nabla \omega _i+f_i g_{X_i} + (d f_i-\omega _i)\otimes e$ and $f_i g_{X_i}, \omega _i$ converges to $f g,\omega$ strongly in $L^2$, respectively by Proposition 3.44 and Proposition 3.70 in [24], we get the lemma. $\square$

We show basic properties about the eigenvalues of ${\bar{\Delta }}^E$ on limit spaces.

Theorem B.7

Let $X\in \overline{{\mathcal {M}}}$.

• (i) We have

  $$\begin{aligned} 0\le \lambda _1({\bar{\Delta }}^{E})\le \lambda _2({\bar{\Delta }}^{E})\le \cdots \rightarrow \infty . \end{aligned}$$

• (ii) There exists a complete orthonormal system of eigensection $\{T_k\}$ in $L^2(E(X))$, i.e., for all $k\in {\mathbb {Z}}_{>0}$, we have $T_k\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ and

  $$\begin{aligned} {\bar{\Delta }}^{E} T_k= \lambda _k({\bar{\Delta }}^{E}) T_k. \end{aligned}$$

• (iii) For all $\lambda \in {\mathbb {R}}$, we have

  $$\begin{aligned} \begin{aligned} \{T\in {\mathcal {D}}^2({\bar{\Delta }}^E,X): {\bar{\Delta }}^{E} T= \lambda T\}&=\bigoplus _{\lambda _k({\bar{\Delta }}^{E}) =\lambda }{\mathbb {R}}T_k\\&:=\mathop {\mathrm {Span}}\nolimits _{{\mathbb {R}}}\{T_k : k\in {\mathbb {Z}}_{>0} \text { with } \lambda _k({\bar{\Delta }}^E) = \lambda \}. \end{aligned} \end{aligned}$$

Proof

Take a sequence $S_i\in H^{1,2}(E(X))$ ($i\in {\mathbb {N}}$) such that $\Vert S_i\Vert _2=1$ and $\lim _{i\rightarrow \infty }\Vert \nabla ^E S_i\Vert _2^2=\lambda _1({\bar{\Delta }}^E,X)$ hold. Since $\sup _{i}\Vert \nabla ^E S_i\Vert _2<\infty$, there exist a subsequence (denote it again by $S_i$) and $T_1\in H^{1,2}(E(X))$ such that $S_i$ converges to $T_1$ strongly in $L^2$ and $\nabla ^E S_i$ converges to $\nabla ^E T_1$ weakly in $L^2$ by the weak compactness of the closed ball in Hilbert spaces and the Rellich theorem for limit spaces [24, Theorem 4.9]. Then, we have $\Vert T_1\Vert _2=1$ and $\Vert \nabla ^E T_1\Vert _2^2\le \liminf \Vert \nabla ^E S_i\Vert _2^2=\lambda _1({\bar{\Delta }}^E,X)$. By the definition of $\lambda _1({\bar{\Delta }}^E,X)$, we have $\Vert \nabla ^E T_1\Vert _2^2\ge \lambda _1({\bar{\Delta }}^E,X)$, and so $\Vert \nabla ^E T_1\Vert _2^2=\lambda _1({\bar{\Delta }}^E,X)$.

For all $t\in {\mathbb {R}}$ and $S\in H^{1,2}(E(X))$ with $\int _X \langle T_1,S\rangle \,d H^n=0$, we have

$$\begin{aligned} \begin{aligned} \lambda _1({\bar{\Delta }}^E,X)(1+t^2\Vert S\Vert _2^2)&=\lambda _1({\bar{\Delta }}^E,X)\Vert T_1+t S\Vert _2^2\\&\le \Vert \nabla ^E T_1+t \nabla ^E S\Vert _2^2\\&= \lambda _1({\bar{\Delta }}^E,X)+\frac{2t}{H^n(X)}\int _X\langle \nabla ^E T_1,\nabla ^E S\rangle \,d H^n+t^2 \Vert \nabla ^E S\Vert _2^2. \end{aligned} \end{aligned}$$

Thus, we have $\int _X\langle \nabla ^E T_1,\nabla ^E S\rangle \,d H^n=0$. Combining this and $\int _X\langle \nabla ^E T_1,\nabla ^E T_1\rangle \,d H^n=\lambda _1({\bar{\Delta }}^E,X) \int _X\langle T_1,T_1\rangle \,d H^n$, we get $\int _X\langle \nabla ^E T_1,\nabla ^E S\rangle \,d H^n=\lambda _1({\bar{\Delta }}^E,X) \int _X\langle T_1,S\rangle \,d H^n$ for all $S\in H^{1,2}(E(X))$. Therefore, we get $T_1\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ and ${\bar{\Delta }}^{E} T_1= \lambda _1({\bar{\Delta }}^{E}) T_1$.

Suppose that we have chosen $T_1,\ldots ,T_k\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ ($k\in {\mathbb {Z}}_{>0}$) such that $\frac{1}{H^n(X)}\int _X\langle T_i, T_j\rangle \,d H^n=\delta _{i j}$ and ${\bar{\Delta }}^{E} T_i= \lambda _i({\bar{\Delta }}^{E}) T_i$ hold. Considering

$$\begin{aligned} \begin{aligned}&{\widetilde{\lambda }}_{k+1}:=\inf \Big \{\Vert \nabla ^E S\Vert _2^2: S\in H^{1,2}_E(E(X))\text { with } \Vert S\Vert _2=1 \text { and}\\&\quad \int _X\langle S,T_i \rangle \,dH^n=0 \hbox { for all }\ i=1,\ldots ,k\Big \}, \end{aligned} \end{aligned}$$

we can take $T_{k+1}\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ such that $\Vert T_{k+1}\Vert _2=1$, ${\bar{\Delta }}^{E} T_{k+1}= {\widetilde{\lambda }}_{k+1} T_{k+1}$ and $\int _X\langle T_{k+1},T_i \rangle \,dH^n=0$ for all $i=1,\ldots ,k$.

Let us show ${\widetilde{\lambda }}_{k+1}=\lambda _{k+1}({\bar{\Delta }}^{E})$. Define $\widetilde{{\mathcal {E}}}_{i}:=\mathop {\mathrm {Span}}\nolimits _{\mathbb {R}}\{T_1,\ldots ,T_{i},T_{k+1}\}$ for $i=1,\ldots ,k$. If ${\widetilde{\lambda }}_{k+1}\ge \lambda _{i}({\bar{\Delta }}^{E})$, we get

$$\begin{aligned} {\widetilde{\lambda }}_{k+1}=\sup _{T\in \widetilde{{\mathcal {E}}}_{i}{\setminus }\{0\}} \frac{\Vert \nabla ^E T\Vert _2^2}{\Vert T\Vert _2^2}\ge \lambda _{i+1}({\bar{\Delta }}^{E}). \end{aligned}$$

Since we have ${\widetilde{\lambda }}_{k+1}\ge \lambda _{1}({\bar{\Delta }}^{E})$, we get ${\widetilde{\lambda }}_{k+1}\ge \lambda _{k+1}({\bar{\Delta }}^{E})$ by induction. To prove ${\widetilde{\lambda }}_{k+1}\le \lambda _{k+1}({\bar{\Delta }}^{E})$, we take arbitrary $(k+1)$-dimensional subspace ${\mathcal {E}}_{k+1}\subset H^{1,2}_E(E(X))$. Then, there exists an element $S\in {\mathcal {E}}_{k+1}$ with $\Vert S\Vert _2=1$ and $\int _X\langle S,T_i \rangle \,dH^n=0$ for all $i=1,\ldots ,k$. Thus, we get

$$\begin{aligned} \sup _{T\in {\mathcal {E}}_{k+1}{\setminus }\{0\}} \frac{\Vert \nabla ^E T\Vert _2^2}{\Vert T\Vert _2^2}\ge {\widetilde{\lambda }}_{k+1}. \end{aligned}$$

This shows that ${\widetilde{\lambda }}_{k+1}\le \lambda _{k+1}({\bar{\Delta }}^{E})$. Therefore, ${\widetilde{\lambda }}_{k+1}= \lambda _{k+1}({\bar{\Delta }}^{E})$.

By induction, we get a sequence $\{T_k\}_{k\in {\mathbb {Z}}_{>0}}\subset {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ such that ${\bar{\Delta }}^{E} T_k= \lambda _k({\bar{\Delta }}^{E}) T_k$ and $\frac{1}{H^n(X)}\int _X\langle T_i, T_j\rangle \,d H^n=\delta _{i j}$.

We next prove that $\lim _{k\rightarrow \infty }\lambda _k({\bar{\Delta }}^{E})=\infty$. Suppose that $\lim _{k\rightarrow \infty }\lambda _k({\bar{\Delta }}^{E})=\lambda <\infty$. Since $\sup _{i}\Vert \nabla ^E T_i\Vert _2\le \lambda$, there exist a subsequence i(j) and $T\in H^{1,2}(E(X))$ such that $T_{i(j)}$ converges to T strongly in $L^2$ and $\nabla ^E T_{i(j)}$ converges to $\nabla ^E T$ weakly in $L^2$. We have $\Vert T\Vert _2=1$. For all $k\in {\mathbb {Z}}_{>0}$, we have

$$\begin{aligned} \int _X\langle T_k, T \rangle \,d H^n=\lim _{l\rightarrow \infty }\int _X\langle T_k, T_l \rangle \,d H^n=0. \end{aligned}$$

Thus, we get

$$\begin{aligned} \int _X\langle T, T \rangle \,d H^n=\lim _{k\rightarrow \infty }\int _X\langle T_k, T \rangle \,d H^n=0. \end{aligned}$$

This contradicts to $\Vert T\Vert _2=1$. Therefore, we get $\lim _{k\rightarrow \infty }\lambda _k({\bar{\Delta }}^{E})=\infty$ and (i).

Suppose that $\bigoplus _{k=1}^\infty {\mathbb {R}} T_k$ is not dense in $H^{1,2}_E(E(X))$. Then, there exists $T\in H^{1,2}_E(E(X))$ such that $\Vert T\Vert _2=1$ and $\int _X\langle T_k, T \rangle \,d H^n=0$ for all $k\in {\mathbb {Z}}_{>0}$. By the definition of ${\widetilde{\lambda }}_k$, we get $\Vert \nabla ^E T\Vert _2^2\ge {\widetilde{\lambda }}_k= \lambda _k({\bar{\Delta }}^{E})$ for all $k\in {\mathbb {Z}}_{>0}$. This contradicts to $\lim _{k\rightarrow \infty }\lambda _k({\bar{\Delta }}^{E})=\infty$. Since $H^{1,2}_E(E(X))$ is dense in $L^2(E(X))$ (see [25, Claim 3.2]), we get that $\{T_k\}$ is complete orthonormal system in $L^2(E(X))$. Thus, we get (ii).

Similarly, we have (iii). $\square$

Let us give the $L^\infty$ estimate for the eigensections of the Laplacian ${\bar{\Delta }}^E$.

Lemma B.8

Let $(M,g)\in {\mathcal {M}}$. Take positive real numbers $\beta >0$ and $0\le \alpha \le \beta$. Then, for any $T\in \Gamma (E(M))$ with ${\bar{\Delta }}^E T=\alpha T$, we have

$$\begin{aligned} \Vert T\Vert _{\infty }\le C(n,K_1,D,\beta ) \Vert T\Vert _2. \end{aligned}$$

Proof

Since we have

$$\begin{aligned} \Delta |T|^2=2\langle {\bar{\Delta }}^E T, T\rangle -2|\nabla ^E T|^2\le 2 \beta |T|^2, \end{aligned}$$

we get the lemma by [32, Proposition 9.2.7] (see also Proposition 7.1.13 and Proposition 7.1.17 in [32]). Note that our sign convention of the Laplacian is different from [32]. $\square$

We investigate the convergence of eigensections. The following proposition plays an important role to prove Theorem B.10.

Proposition B.9

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be a sequence in ${\mathcal {M}}$ and let $X\in \overline{{\mathcal {M}}}$ be the Gromov–Hausdorff limit. Take a sequence $\{\lambda _i\}$ in ${\mathbb {R}}_{\ge 0}$ and $T_i\in \Gamma (E(X_i))$ with ${\bar{\Delta }}^{E} T_i =\lambda _i T_i$ and $\Vert T_i\Vert _2=1$. Suppose that $T_i$ converges to $T\in L^2(E(X))$ weakly in $L^2$, and there exists $\mu >0$ with $\sup _{i}\lambda \le \mu$. Then, we have the following properties.

• (i) There exist $\lambda \in {\mathbb {R}}_{\ge 0}$ with $\lambda =\lim _{i\rightarrow \infty } \lambda _i$.

• (ii) We have $T\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)\cap L^\infty (E(X))$, $\Vert T\Vert _{\infty }\le C(n,K,D,\mu )$ and ${\bar{\Delta }}^E T=\lambda T$.

• (iii) $T_i,\nabla ^E T_i$ converges $T,\nabla ^E T$ strongly in $L^2$, respectively.

Proof

Take $\omega _i\in \Gamma (T^*X_i)$ and $f_i\in C^\infty (X_i)$ with $T_i=\omega _i+f_i e$. Since $\Vert T_i\Vert _{\infty }\le C(n,K,D,\beta )$, we have $\Vert \omega _i\Vert _{\infty } \le C(n,K,D,\beta )$. By [24, Theorem 4.9] and [26, Theorem 6.11] (see also [25, Proposition 4.2]), there exist a subsequence i(j), $\omega \in L^2(T^*X)$ and $f\in H^{1,2}(X)$ such that $\omega$ is differentiable at almost all point in X, $\nabla \omega \in L^2(T^*X\otimes T^*X)$, $\omega _i$, $f_i$ converges to $\omega$, f strongly in $L^2$ and $\nabla \omega _i$, $d f_i$ converges to $\nabla \omega$, df weakly in $L^2$, respectively. By the lower semi-continuity of $L^\infty$ norm [24, Proposition 3.64], we have $\Vert f\Vert _{\infty }\le C(n,K,D,\mu )$, $\Vert \omega \Vert _{\infty }\le C(n,K,D,\mu )$ and $\Vert T\Vert _{\infty }\le C(n,K,D,\mu )$. As with the proof of [25, Proposition 4.8 (ii)], we have $\langle \omega ,d h\rangle \in H^{1,2}(X)$ for all $h\in {\mathcal {D}}^2(\Delta ,X)$ with $\Delta h\in L^{\infty }(X)$. By [25, Proposition 4.5], we get $\omega \in H^{1,2}_C(T^*X)$.

Take arbitrary $S=\eta +he\in \mathop {\mathrm {Test}}\nolimits E(X)$. Then, there exist sequences $\eta _i\in \Gamma (T^*X_i)$ and $h_i\in C^\infty (X_i)$ such that $\eta _i$, $\nabla \eta _i$, $\delta \eta _i$, $h_i$, $d h_i$ converges to $\eta$, $\nabla \eta$, $\delta \eta$, h, dh strongly in $L^2$, respectively by [25, Proposition 3.5]. Set $S_i:= \eta _i+ h_i e\in \Gamma (E(X_i))$. Take a subsequence i(j) such that the limit $\lambda =\lim _{j\rightarrow \infty } \lambda _{i(j)}$ exists. Then, we get

$$\begin{aligned} \begin{aligned} \lim _{j\rightarrow \infty }\int _{X_{i(j)}}\langle \nabla ^E T_{i(j)},\nabla ^E S_{i(j)}\rangle \,d H^n&= \lim _{j\rightarrow \infty }\lambda _{i(j)}\int _{X_{i(j)}}\langle T_{i(j)}, S_{i(j)}\rangle \,d H^n\\&=\lambda \int _{X}\langle T, S \rangle \,d H^n. \end{aligned} \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned}&\lim _{j\rightarrow \infty }\int _{X_{i(j)}}\langle \nabla ^E T_{i(j)},\nabla ^E S_{i(j)}\rangle \,d H^n\\&\quad =\lim _{j\rightarrow \infty }\int _{X_{i(j)}}\Big ( \langle \nabla \omega _{i(j)},\nabla \eta _{i(j)}\rangle -f_{i(j)}\delta \eta _{i(j)} - h_{i(j)}\delta \omega _{i(j)} +n f_{i(j)} h_{i(j)}\\&\qquad +\langle d f_{i(j)}-\omega _{i(j)},d h_{i(j)}-\eta _{i(j)}\rangle \Big ) \,d H^n\\&\quad = \int _{X}\Big ( \langle \nabla \omega ,\nabla \eta \rangle -f\delta \eta - h\delta \omega +n f h +\langle d f-\omega ,d h-\eta \rangle \Big ) \,d H^n\\&\quad =\int _{X}\langle \nabla ^E T,\nabla ^E S\rangle \,d H^n. \end{aligned} \end{aligned}$$

Thus, we get

$$\begin{aligned} \int _{X}\langle \nabla ^E T,\nabla ^E S\rangle \,d H^n=\lambda \int _{X}\langle T, S \rangle \,d H^n. \end{aligned}$$

This shows that $T\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)\cap L^\infty (E(X))$ and ${\bar{\Delta }}^E T=\lambda T$. Since $\lambda$ is uniquely determined, we have $\lambda =\lim _{i\rightarrow \infty } \lambda _i$. Thus, we get (i) and (ii)

Since we have

$$\begin{aligned} \lim _{i\rightarrow \infty }\Vert \nabla ^E T_i\Vert _2^2=\lim _{i\rightarrow \infty }\lambda _i=\lambda =\Vert \nabla ^E T\Vert _2^2, \end{aligned}$$

$\nabla ^E T_i$ converges to $\nabla ^E T$ strongly in $L^2$. $\square$

Let us show the continuity of the eigenvalues of the Laplacian ${\bar{\Delta }}^E$ under our setting. The following theorem is the main goal of this appendix.

Theorem B.10

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be a sequence in $\overline{{\mathcal {M}}}$ and let $X\in \overline{{\mathcal {M}}}$ be the Gromov–Hausdorff limit. Then, we have

$$\begin{aligned} \lim _{i\rightarrow \infty }\lambda _k({\bar{\Delta }}^E,X_i)=\lambda _k({\bar{\Delta }}^E,X) \end{aligned}$$

for all $k\in {\mathbb {Z}}_{>0}$.

Proof

We consider the case when $X_i\in {\mathcal {M}}$ for all i. The general case is its easy consequence.

Take arbitrary $k\in {\mathbb {Z}}_{>0}$.

We first show the lower semi-continuity

$$\begin{aligned} \liminf _{i\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_i)\ge \lambda _k({\bar{\Delta }}^E,X). \end{aligned}$$

If $\liminf _{i\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_i)=\infty$, this is trivial. We assume that $\liminf _{i\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_i)<\infty$. For each i, let $\{T_{i,j}\}_{j\in {\mathbb {Z}}_{>0}}$ be the complete orthonormal system of eigensections in $L^2(E(X_i))$:

$$\begin{aligned} {\bar{\Delta }}^E T_{i,j}=\lambda _j({\bar{\Delta }}^E,X_i)T_{i,j}. \end{aligned}$$

We can take a subsequence i(l) such that $\liminf _{i\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_i)=\lim _{l\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_{i(l)})$ and $T_{i(l),j}$ converges to some $T_j\in L^2(E(X))$ weakly in $L^2$ as $l\rightarrow \infty$ for all $j=1,\ldots ,k$ by [24, Proposition 3.50]. By Proposition B.9, there exist $\lambda _1,\ldots ,\lambda _k$ with $\lambda _j=\lim _{l\rightarrow \infty }\lambda _j({\bar{\Delta }}^E,X_{i(l)})$ for all $j=1,\ldots ,k$, and we have $T_j\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ and ${\bar{\Delta }}^E T_{j}=\lambda _j T_{j}$. Define ${\mathcal {E}}_{k}:=\mathop {\mathrm {Span}}\nolimits _{{\mathbb {R}}}\{T_1,\ldots ,T_k\}$. Then, we get

$$\begin{aligned} \lambda _k({\bar{\Delta }}^E,X)\le \sup _{T\in {\mathcal {E}}_{k}{\setminus }\{0\}}\frac{\Vert \nabla ^E T\Vert _2^2}{\Vert T\Vert _2^2}=\liminf _{i\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_i). \end{aligned}$$

We next show the upper semi-continuity

$$\begin{aligned} \limsup _{i\rightarrow \infty } \lambda _k({\bar{\Delta }}^E,X_i)\le \lambda _k({\bar{\Delta }}^E,X). \end{aligned}$$

Let $\{T_{j}\}_{j\in {\mathbb {Z}}_{>0}}$ be the complete orthonormal system of eigensections in $L^2(E(X))$:

$$\begin{aligned} {\bar{\Delta }}^E T_{j}=\lambda _j({\bar{\Delta }}^E,X)T_{j}. \end{aligned}$$

For each j, take $\omega _j\in H^{1,2}_C(T^*X)$ and $f_j\in H^{1,2}(X)$ such that $T_j=\omega _j+f_j e$. By [25, Theorem 3.5], there exist $\omega _{i,j}\in \Gamma (T^*X_i)$ and $f_{i,j}\in C^\infty (X_i)$ such that $\omega _{i,j}$, $\nabla \omega _{i,j}$, $f_{i,j}$, $d f_{i,j}$ converges to $\omega _j$, $\nabla \omega _{j}$, $f_{j}$, $d f_{j}$ strongly in $L^2$, respectively. Define $T_{i,j}:=\omega _{i,j}+f_{i,j}e\in \Gamma (E(X_i))$ and ${\mathcal {E}}_{i,k}:=\mathop {\mathrm {Span}}\nolimits _{{\mathbb {R}}}\{T_{i,1},\ldots ,T_{i,k}\}\subset \Gamma (E(X_i))$. Then, $\dim {\mathcal {E}}_{i,k}=k$ for sufficient large i, and

$$\begin{aligned} \lambda _k({\bar{\Delta }}^E,X)=\lim _{i\rightarrow \infty }\sup _{T\in {\mathcal {E}}_{i,k}{\setminus }\{0\}}\frac{\Vert \nabla ^E T\Vert _2^2}{\Vert T\Vert _2^2}. \end{aligned}$$

Thus, we get

$$\begin{aligned} \limsup _{i\rightarrow \infty }\lambda _k({\bar{\Delta }}^E,X_i)\le \limsup _{i\rightarrow \infty }\sup _{T\in {\mathcal {E}}_{i,k}{\setminus }\{0\}}\frac{\Vert \nabla ^E T\Vert _2^2}{\Vert T\Vert _2^2}=\lambda _k({\bar{\Delta }}^E,X). \end{aligned}$$

$\square$

Let us investigate the relationship between $\lambda _k({\bar{\Delta }}^E)$ and the Obata equation $\nabla ^2 f + f g=0$.

Theorem B.11

Let $X\in \overline{{\mathcal {M}}}$. For all $k\in {\mathbb {Z}}_{>0}$ the following conditions are mutually equivalent.

• (i) $\lambda _k({\bar{\Delta }}^E,X)=0$.

• (ii) There exists a k-dimensional subspace $V\subset {\mathcal {D}}^2(\Delta ,X)$ such that $\nabla ^2 f+f g=0$ holds for all $f\in V$.

Proof

We first prove that (ii) implies (i). For all $f\in {\mathcal {D}}^2(\Delta ,X)$, we have $d f\in H^{1,2}_C(T^*X)$ (see [26, Theorem 1.9] and the definition of $H^{1,2}_C(T^*X)$), and so $d f + f e\in H^{1,2}_E(E(X))$ by Lemma B.4. Define ${\widetilde{V}}:=\{d f + f e: f\in V\}\subset H^{1,2}_E(E(X))$. Since $\nabla ^E T=0$ for all $T\in {\widetilde{V}}$, we have ${\widetilde{V}}\subset \{T\in {\mathcal {D}}^2({\bar{\Delta }}^E,X):{\bar{\Delta }}^E T=0\}$. Thus, we get $\lambda _k({\bar{\Delta }}^E,X)=0$ by Theorem B.7.

We next prove that (i) implies (ii). For all $T=\omega +f e\in {\mathcal {D}}^2({\bar{\Delta }}^E,X)$ with ${\bar{\Delta }}^E T=0$, we have $f\in {\mathcal {D}}^2(\Delta ,X)$ (see Lemma B.5), $\nabla \omega +f g=0$ and $d f-\omega =0$, and so $\nabla ^2 f+ fg =0$. Since the map $\{T\in {\mathcal {D}}^2({\bar{\Delta }}^E,X):{\bar{\Delta }}^E T=0\}\rightarrow {\mathcal {D}}^2(\Delta ,X),\, T=\omega +fe \mapsto f$ is injective, the image $V\subset {\mathcal {D}}^2(\Delta ,X)$ satisfies $\dim V\ge k$. Thus, we get (ii). $\square$

Corollary B.12

Let $X\in \overline{{\mathcal {M}}}$. If there exists a non-zero element $f\in {\mathcal {D}}^2(\Delta ,X)$ with $\nabla ^2 f+f g=0$, then there exists a compact geodesic space Z such that X is isometric to the spherical suspension $S^0*Z$.

Proof

By Theorem B.11, we have $\lambda _1({\bar{\Delta }}^E,X)=0$. Take a sequence $\{X_i\}$ in ${\mathcal {M}}$ with $X_i$ converges to X in the Gromov–Hausdorff topology. Then, we have $\lim _{i\rightarrow \infty }\lambda _1({\bar{\Delta }}^E,X_i)=0$. Thus, we get the corollary by Theorem A.1. $\square$

We get the following three corollaries immediately by Theorems A.1, B.9 and Main Theorem 1.

Corollary B.13

Given a positive real number $\epsilon >0$ and a integer $1\le k\le n-1$, there exists $\delta (n,K_1,D,v,\epsilon )>0$ such that if $X\in \overline{{\mathcal {M}}}$ satisfies $\lambda _k({\bar{\Delta }}^E,X)\le \delta$, then $d_{GH}(X,S^{k-1}*Z)\le \epsilon$ for some compact geodesic space Z.

Corollary B.14

Let $\{X_i\}_{i\in {\mathbb {N}}}$ be a sequence in $\overline{{\mathcal {M}}}$. Then the following three conditions are mutually equivalent.

• (i) $\lambda _n({\bar{\Delta }}^E,X_i)\rightarrow 0$ as $i\rightarrow \infty$.

• (ii) $\lambda _{n+1}({\bar{\Delta }}^E,X_i)\rightarrow 0$ as $i\rightarrow \infty$.

• (iii) $d_{GH}(X_i,S^n)\rightarrow 0$ as $i\rightarrow \infty$.

Corollary B.15

Given a positive real number $\epsilon >0$, there exists $\delta (n,K_1,D,\epsilon )>0$ such that if $X\in \overline{{\mathcal {M}}}$ satisfies $\lambda _n({\bar{\Delta }}^E,X)\le \delta$, then $d_{GH}(X,S^n)\le \epsilon$.

Note that $\delta$ in Corollary B.15 does not depend on $K_2$ and v. If $\epsilon$ is sufficiently small in Corollary B.15, then X is bi-Hölder equivalent to $S^n$ by Theorem 5.9, Theorem A.1.2, Theorem A.1.3, Theorem A.1.5 in [8]. In particular, X is homeomorphic to $S^n$.