Proof. We give only a proof of (1) because (2) is similar. Denoting by $\tilde{\mathfrak {m}},\, \tilde x$ the reference measure, the base point, respectively on a tangent cone, it easily follows from the finiteness $\nu _x<\infty$ that

(2.29)\begin{equation} \frac{\tilde{\mathfrak{m}} (B_R(\tilde x))}{\tilde{\mathfrak{m}} (B_r(\tilde x))}=\left(\frac{R}{r}\right)^N, \quad \forall r>0, \quad \forall R>0. \end{equation}

Thus (2) of theorem 2.8 allows us to conclude.

Proof. It follows by definitions 2.5 and 2.7 that (2.30) holds. On the other hand, (2.31) is a direct consequence of [Reference Kuwada and Li45, Proposition 4.10] with (2.11) and (2) of theorem 2.8. See [Reference Huang41, Proposition 2.13] for a more general result (see also [Reference Ding23, Theorem 6.20] and [Reference Taylor55, Section 8]).

Proof. We only show (3.6) under assuming $\nu _x<\infty$ because the others can follow from similar arguments. The Bishop–Gromov inequality (2.16) shows that the map

(3.10)\begin{equation} I_x(r):=\nu_x-\frac{\mathfrak{m}(B_r(x))}{r^N} \end{equation}

is non-decreasing with $I_x(r)\geqslant 0$. Thus fixing $r_0>0$, we have for any $0< r< r_0$,

(3.11)\begin{equation} \nu_x r^N-I_x(r_0)r^N\leqslant\mathfrak{m}(B_r(x))\leqslant \nu_x r^N. \end{equation}

Write $F_x(r)$ as

(3.12)\begin{equation} F_x(r)=\int_{r}^{r_0}\frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s+\int_{r_0}^\infty\frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s. \end{equation}

Then by (3.11) the first term of the right-hand-side can be estimated as follows.

(3.13)\begin{equation} \frac{r_0^{2-N}-r^{2-N}}{(2-N)\nu_x}\leqslant \int_r^{r_0} \frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s\leqslant \frac{r_0^{2-N}-r^{2-N}}{(2-N)(\nu_x-I_x(r_0))}. \end{equation}

Thus

(3.14)\begin{align} & r^{N-2}\int_{r_0}^\infty\frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s-C(N,\nu_x)\left(\frac{r}{r_0}\right)^{N-2}\leqslant\frac{F_x(r)}{r^{2-N}}-\frac{1}{(N-2)\nu_x}\nonumber\\ & \quad \leqslant r^{N-2}\int_{r_0}^\infty\frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s+\frac{C(N,\nu_x)}{I_x(r_0)}\left(\frac{r}{r_0}\right)^{N-2}. \end{align}

For any $\varepsilon \in (0,\,1)$, we can let $r=\varepsilon r_0$ and thus

(3.15)\begin{align} (\varepsilon r_0)^{N-2}\int_{r_0}^\infty& \frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s-C(N,\nu_x)\varepsilon^{N-2}\leqslant\frac{F_x(\varepsilon r_0)}{(\varepsilon r_0)^{2-N}}-\frac{1}{(N-2)\nu_x}\nonumber\\ & \leqslant (\varepsilon r_0)^{N-2}\int_{r_0}^\infty\frac{s}{\mathfrak{m}(B_s(x))}\, \mathrm{d} s+\frac{C(N,\nu_x)}{I_x(r_0)}\varepsilon^{N-2}. \end{align}

Letting $\varepsilon \rightarrow 0$ completes the proof of (3.6).

Proof. Because

(3.21)\begin{equation} \Delta \mathsf{b}_x=\Delta G_x^{\frac{1}{2-N}}=\frac{N-1}{(N-2)^2}G_x^{\frac{2N-3}{2-N}}|\nabla G_x|^2=(N-1)\frac{|\nabla \mathsf{b}_x|^2}{\mathsf{b}_x}, \end{equation}

and thus

(3.22)\begin{equation} \Delta \mathsf{b}_x^2=2\mathsf{b}_x\Delta \mathsf{b}_x+2|\nabla \mathsf{b}_x|^2= 2N|\nabla \mathsf{b}_x|^2, \end{equation}

where we used the fact that $G_x$ is harmonic on $X \setminus \{x\}$.

Proof. Thanks to proposition 2.10, we know

(3.26)\begin{align} G_x(y)& =\frac{2^{1-N}}{N\Gamma \left(\frac{N}{2}\right)\mathfrak{m} (B_1(x))}\int_0^{\infty}t^{-\frac{N}{2}}\exp \left(-\frac{\mathsf{d}(x, y)^2}{4t}\right)\mathrm{d} t \nonumber\\ & =\frac{2^{1-N}}{N\Gamma \left(\frac{N}{2}\right)\mathfrak{m} (B_1(x))} \cdot \mathsf{d}(x,y)^{2-N}4^{\frac{N}{2}-1}\Gamma \left(\frac{N}{2}-1\right) \nonumber\\ & =\frac{1}{N(N-2)\mathfrak{m} (B_1(x))}\mathsf{d}(x, y)^{2-N}. \end{align}

Finally recalling (2.30), we get (3.24).

Proof. The formula for $G$ is a direct consequence of (2.13). Moreover it implies (3.29). On the other hand, since

(3.32)\begin{align} F_{\tilde x}(r)& =\int_r^{\infty}\frac{s}{b\mathfrak{m}(\tilde B_s(\tilde x))}\mathrm{d} s=\frac{1}{b}\int_r^{\infty}\frac{s}{\mathfrak{m} (B_{a^{{-}1}s}(x))}\mathrm{d} s \nonumber\\ & =\frac{a^2}{b}\int_{a^{{-}1}r}^{\infty}\frac{t}{\mathfrak{m} (B_t(x))}\mathrm{d} t=\frac{a^2}{b}F_x\left(\frac{r}{a}\right), \end{align}

we have the desired formula for $F$. Similarly we have the remaining results.

Proof. We give only a proof of (3.35) because the proof (3.34) is similar (moreover this is valid even in the case of negative lower bounds on Ricci curvature. See also [Reference Lytchak and Stadler47, Subsection 2.3]). For fixed $r>0$, we have

(3.36)\begin{equation} V_{X_i} \le \frac{\mathfrak{m}_i(B_r(x_i))}{r^N} \to \frac{\mathfrak{m} (B_r(x))}{r^N} \end{equation}

which shows

(3.37)\begin{equation} \limsup_{i \to \infty}V_{X_i} \le \frac{\mathfrak{m}(B_r(x))}{r^N}. \end{equation}

Then letting $r \to \infty$ completes the proof of (3.35).

Proof. Note that (3.41) gives a uniform positive lower (upper, respectively) bound on $\mathfrak {m}_i(B_1(x_i))$ because of (3.39). Thus, thanks to the compactness of $\operatorname {RCD}$ spaces with respect to the pmGH topology (see for instance [Reference Ambrosio, Gigli and Savaré1, Theorem 6.11], [Reference Erbar, Kuwada and Sturm25, Theorem 5.3.22], [Reference Gigli, Mondino and Savaré28, Theorem 7.2], [Reference Lott and Villani46, Theorem 5.19], and [Reference Sturm53, Theorem 4.20]), after passing to a subsequence, $(X_i,\,\mathsf {d}_i,\,\mathfrak {m}_i,\,x_i)$ pmGH converge to a pointed $\operatorname {RCD}(0,\, N)$ space $(X,\,\mathsf {d},\,\mathfrak {m},\,x)$. Observe that for all $r \leqslant s$, we have

(3.43)\begin{equation} \int_{r_i}^s\frac r{\mathfrak{m}_i(B_t(x_i))}\, \mathrm{d} t\rightarrow\int_r^s\frac{t}{\mathfrak{m}(B_{t}(x))}\, \mathrm{d} t, \end{equation}

which implies

(3.44)\begin{equation} \int_r^s\frac{t}{\mathfrak{m}(B_{t}(x))}\, \mathrm{d} t= \lim_{i \to \infty}\int_{r_i}^s\frac t{\mathfrak{m}_i(B_t(x_i))}\, \mathrm{d} t \leqslant \liminf_{i \to \infty}F_{x_i}(r_i). \end{equation}

Letting $s\rightarrow \infty$, we have (3.42).

Proof. The key point is:

• • it is proved in [Reference Ambrosio, Honda and Tewodrose4, Theorem 3.3] that

  (3.48)\begin{equation} p_{X_i}(y_i,z_i,t_i)\to p_X(y,z,t) \end{equation}
  holds for all convergent sequences of $t_i \to t$ in $(0,\, \infty )$ and of $y_i,\, z_i \in X_i \to y, z \in X$, respectively.

Based on the above, let us start giving the proof. The implication from (1) to (2) is trivial. Assume that (2) holds. Thanks to (3.48) and (2.20) with the assumption, we know $\int _r^Rp_i(x_i,\, y_i,\,t)\mathrm {d} t \to \int _r^Rp(x,\, y,\, t)\mathrm {d} t$ and thus $F_{x_i}(R) \to F_x(R)$ for any $R \geqslant 1$. In particular for any $0<\varepsilon <1$ there exists $R\geqslant 1$ such that $F_{x_i}(R) + F_x(R)<\varepsilon$ for any $i$. On the other hand, for any fixed convergent sequence $y_i \in X_i$ to $y \in X$ with $x \neq y$, by (2.20), we know that there exists $0< r<1$ such that

(3.49)\begin{equation} \int_0^rp_i(x_i, y_i, t)\mathrm{d} t+\int_0^rp(x, y, t)\mathrm{d} t<\varepsilon. \end{equation}

The observation above allows us to conclude the pointwise convergence $G_{x_i}(y_i) \to G_x(y)$. Then the locally uniform convergence comes from this with a locally uniform Lipschitz bound (3.17). Moreover since $G_{x_i}$ is harmonic on $X_i\setminus \{x_i\}$, it follows from the stability of Laplacian, [Reference Ambrosio and Honda3, Theorem 4.4], that the $W^{1,2}_{\mathrm {loc}}$-strong convergence of the Green functions holds. Finally the improvement to the $W^{1,p}_{\mathrm {loc}}$-strong convergence is justified by combining this with (3.17) (see also [Reference Honda39]). Thus we have (3).

Finally let us prove the remaining implication from (3) to (1). Thanks to corollary 3.5, for any $0<\varepsilon <1$ there exists $R\geqslant 1$ such that $G_x(y)<\varepsilon$ for any $y \in X \setminus B_{R}(x)$. Fix $y \in X \setminus B_{2R}(x)$ and take $y_i \in X_i$ converging to $y$. Then our assumption allows us to conclude $G_{x_i}(y_i)<2\varepsilon$ for any sufficiently large $i$. Thus by (3.16), we have

(3.50)\begin{equation} F_{x_i}(R) \leqslant F_{x_i}(\mathsf{d}_i(x_i, y_i))\leqslant C(N) G_{x_i}(y_i)\leqslant C(N)\varepsilon. \end{equation}

On the other hand, as discussed above, we can prove that $f_i$ converges in $L^1([1,\, r),\, \mathscr {H}^1)$ to $f$ for any finite $r>1$. This with (3.50) implies (1) because $\varepsilon$ is arbitrary.

Proof. We prove only (3.51) via a blow-up argument because the proof of (3.52) is similar via a blow-down argument, where the case when $\nu _x=\infty$ or $V_X=0$ directly follows from lemma 3.3 with (3.16).

Take a convergent sequence $y_i \in X \setminus \{x\} \to x$, let $r_i:=\mathsf {d}(x,\, y_i)$ and consider rescaled $\operatorname {RCD}(0,\, N)$ spaces;

(3.53)\begin{equation} (X_i, \mathsf{d}_i, \mathfrak{m}_i, x_i):=\left( X, \frac{1}{r_i}\mathsf{d}, \frac{1}{r_i^N}\mathfrak{m}, x\right). \end{equation}

Then since

(3.54)\begin{equation} F_{x_i}(1)=\frac{F_x(r_i)}{r_i^{2-N}} \to \frac{1}{(N-2)\nu_x}, \quad \nu_{x_i}=\nu_x, \end{equation}

thanks to theorem 3.11, after passing to a subsequence, $(X_i,\, \mathsf {d}_i,\, \mathfrak {m}_i,\, x_i)$ pmGH converge to a tangent cone $(W,\, \mathsf {d}_W,\, \mathfrak {m}_W,\, w)$ at $x$, which is isomorphic to the non-parabolic $N$-metric measure cone over an $\operatorname {RCD}(N-2,\, N-1)$ space with the finite volume density $\nu _w=\nu _x$ (see (2.30)). By proposition 3.8 we know $F_w(1)=\frac {1}{(N-2)\nu _w} (=\lim _{i \to \infty }F_{x_i}(1)$ by (3.54)) and

(3.55)\begin{equation} G^W(w,z)=\frac{1}{N(N-2)\nu_w} \mathsf{d}_W(w, z)^{2-N}, \quad \forall z \in W \setminus \{w\}. \end{equation}

After passing to a subsequence again, we find the limit point $z$ of $y_i \in X_i$ (thus $\mathsf {d}_W(w,\, z)=1$). Then theorem 3.12 shows

(3.56)\begin{equation} \frac{G(x, y_i)}{r_i^{2-N}}=G^{X_i}(x_i, y_i) \to G^Y(w, z)= \frac{1}{N(N-2)\nu_w}=\frac{1}{N(N-2)\nu_x} \end{equation}

which completes the proof because $y_i$ is arbitrary.

Proof. In order to simplify our arguments below, we give a proof only in the case when $p=2$ because the general case is similar after replacing $W^{1,2}$-convergence by $W^{1, p_i}$-convergence for a convergent sequence $p_i \to p$.

The proof is done by a strandard contradiction argument based on the compactness of $\operatorname {RCD}$ spaces with respect to the pmGH convergence. Namely if the assertion is not satisfied, then there exist sequences of;

• • pointed non-parabolic $\operatorname {RCD}(0,\, N)$ spaces $(X_{j, i},\, \mathsf {d}_{j, i},\, \mathfrak {m}_{j, i},\, x_{j, i})$ with $\mathfrak {m}_{j,i}(B_1 (x_{j, i}))\leqslant v$,

  (3.60)\begin{equation} f_{j, i}(s):=\frac{s}{\mathfrak{m}_{j, i}(B_s(x_{j, i}))}\leqslant \varphi(s),\quad \text{for }\mathscr{H}^1\text{-a.e. }s \in [1, \infty) \end{equation}
  and
  (3.61)\begin{equation} \mathsf{d}_{\mathrm{pmGH}}\left((X_{1, i}, \mathsf{d}_{1, i}, \mathfrak{m}_{1, i}, x_{1, i}), (X_{2, i}, \mathsf{d}_{2, i}, \mathfrak{m}_{2, i}, x_{2, i})\right) \to 0; \end{equation}

• • points $y_{j, i} \in B_R(x_{j, i}) \setminus B_r(x_{j, i})$ satisfying that $y_{1, i}$ is $\varepsilon _i$-close to $y_{2, i}$ for some $\varepsilon _i \to 0^+$ and that

  (3.62)\begin{equation} \inf_i\left( \left| \mathsf{b}_{x_{1, i}}(y_{1, i})-\mathsf{b}_{x_{2, i}}(y_{2, i})\right|+\left|{\unicode{x2A0D}}_{B_s(y_{1, i})}|\nabla \mathsf{b}_{x_{1,i}}|^2\mathrm{d} \mathfrak{m}_{1, i} - {\unicode{x2A0D}}_{B_s(y_{2, i})}|\nabla \mathsf{b}_{x_{2, i}}|^2\mathrm{d} \mathfrak{m}_{2, i}\right|\right)>0. \end{equation}

Theorem 3.11 shows that after passing to a subsequence, $(X_{j, i},\, \mathsf {d}_{j, i},\, \mathfrak {m}_{j, i},\, x_{j, i})$ pmGH-converge to a pointed non-parabolic $\operatorname {RCD}(0,\, N)$ space $(X,\, \mathsf {d},\, \mathfrak {m},\, x)$. With no loss of generality we can assume that $y_{j, i}$ converge to a point $y \in \overline {B}_R(x) \setminus B_r(x)$. Moreover the dominated convergence theorem with (3.60) yields that $f_{j,i}(s)$ $L^1$-strongly converge to $f(s):=\frac {s}{\mathfrak {m} (B_s(x))}$ in $L^1([1,\, \infty ),\, \mathscr {H}^1)$. Thus theorem 3.12 allows us to conclude

(3.63)\begin{align} & \left| \mathsf{b}_{x_{1, i}}(y_{1, i})-\mathsf{b}_{x_{2, i}}(y_{2, i})\right|+\left|{\unicode{x2A0D}}_{B_s(y_{1, i})}|\nabla \mathsf{b}_{x_{1,i}}|^2\mathrm{d} \mathfrak{m}_{1, i} - {\unicode{x2A0D}}_{B_s(y_{2, i})}|\nabla \mathsf{b}_{x_{2, i}}|^2\mathrm{d} \mathfrak{m}_{2, i}\right| \nonumber\\ & \quad \to \left| \mathsf{b}_{x}(y)-\mathsf{b}_{x}(y)\right|+\left|{\unicode{x2A0D}}_{B_s(y)}|\nabla \mathsf{b}_{x}|^2\mathrm{d} \mathfrak{m} - {\unicode{x2A0D}}_{B_s(y)}|\nabla \mathsf{b}_{x}|^2\mathrm{d} \mathfrak{m}\right|=0 \end{align}

which contradicts (3.62).

Proof. By theorem 3.4,

(3.72)\begin{equation} |\nabla \log G_x|=\frac{|\nabla G_x|}{G_x}\leqslant C(N)\frac{H_x(\mathsf{d}(x,\cdot))}{F_x(\mathsf{d}(x,\cdot))}. \end{equation}

On the other hand by definition we have

(3.73)\begin{equation} F_x(s)=\int_s^{\infty}\frac{t}{\mathfrak{m} (B_t(x))}\mathrm{d} t \geqslant \int_s^{\infty}\frac{s}{\mathfrak{m} (B_t(x))}\mathrm{d} t =sH_x(s). \end{equation}

Combining this with (3.72) completes the proof.

Proof. The proof is quite standard;

(3.81)\begin{align} \int_{\Omega}\langle \nabla \varphi, \nabla u\rangle \mathrm{d} \mathfrak{m}_{G_x} & =\int_{\Omega}\langle \nabla (G_x^2\varphi), \nabla u\rangle \mathrm{d} \mathfrak{m} -\int_{\Omega}\varphi \langle \nabla G_x^2, \nabla u\rangle \mathrm{d} \mathfrak{m} \nonumber\\ & ={-}\int_{\Omega}G_x^2\varphi\Delta u\mathrm{d} \mathfrak{m}-\int_{\Omega}2\varphi G_x\langle \nabla G_x,\nabla u\rangle \mathrm{d} \mathfrak{m} \nonumber\\ & ={-}\int_{\Omega} \varphi \cdot \mathscr L u \mathrm{d} \mathfrak{m}_{G_x}. \end{align}

Proof. First of all, we claim that $|\nabla \mathsf {b}_x|^2\in W^{1,2}_{\mathrm {loc}}(X\setminus \{x\},\, \mathsf {d},\, \mathfrak {m})$. For any compact set $K\subset X\setminus \{x\}$, we can take a good cut-off function $\eta \in (\operatorname {Lip}_c)_+(X,\, \mathsf {d})\cap D(\Delta )$ such that $\eta \equiv 1$ in $K$, $\operatorname {supp}\eta \subset X\setminus \{x\}$ and $|\nabla \eta |+|\Delta \eta |< C$ (see [Reference Mondino and Naber48, Lemma 3.2] for such an existence). Letting $h:=\eta \mathsf {b}_x$, then [Reference Gigli34, Propositions 3.3.18 and 3.3.22] shows that $|\nabla h|^2\in W^{1,2}(X\setminus \{x\},\, \mathsf {d},\, \mathfrak {m})$, which implies $|\nabla \mathsf {b}_x|^2\in W^{1,2}_{\mathrm {loc}}(X\setminus \{x\},\, \mathsf {d},\, \mathfrak {m})$ because $K$ is arbitrary.

Next recalling

(3.82)\begin{align} & -\int_{X\setminus \{x\}}\langle \nabla \psi, \nabla (fh)\rangle \mathrm{d} \mathfrak{m}=\int_{X\setminus \{x\}}\psi \left(h\Delta f+2\langle \nabla f, \nabla h\rangle \right)\mathrm{d} \mathfrak{m}\nonumber\\ & \quad -\int_{X\setminus \{x\}}\langle \nabla(\psi f), \nabla h\rangle \mathrm{d} \mathfrak{m} \end{align}

for all $\psi \in \mathrm {Lip}_c(X\setminus \{x\},\, \mathsf {d})$, $f \in D_{\mathrm {loc}}(\Delta,\, X\setminus \{x\})$ and $h \in W^{1,2}_{\mathrm {loc}}(X\setminus \{x\},\, \mathsf {d},\, \mathfrak {m})$, we apply this as $f=\mathsf {b}_x^2,\, h=|\nabla \mathsf {b}_x|^2$ to get

(3.83)\begin{align} & \quad -\int_{X\setminus \{x\}}\langle \nabla \psi, \nabla (\mathsf{b}_x^2|\nabla \mathsf{b}_x|^2)\rangle \mathrm{d} \mathfrak{m} \nonumber\\ & =\int_{X\setminus \{x\}}\psi \left(|\nabla \mathsf{b}_x|^2\Delta \mathsf{b}_x^2+2\langle \nabla \mathsf{b}_x^2, \nabla |\nabla \mathsf{b}_x|^2\rangle \right)\mathrm{d} \mathfrak{m}-\int_{X\setminus \{x\}}\langle \nabla(\psi \mathsf{b}_x^2), \nabla |\nabla \mathsf{b}_x|^2\rangle \mathrm{d} \mathfrak{m} \nonumber\\ & =\int_{X\setminus \{x\}}\psi \left(2N|\nabla \mathsf{b}_x|^4+2\langle \nabla \mathsf{b}_x^2, \nabla |\nabla \mathsf{b}_x|^2\rangle \right)\mathrm{d} \mathfrak{m}-\int_{X\setminus \{x\}}\langle \nabla(\psi \mathsf{b}_x^2), \nabla |\nabla \mathsf{b}_x|^2\rangle \mathrm{d} \mathfrak{m}, \end{align}

where we used (3.20). On the other hand, since $\mathsf {b}_x^2|\nabla \mathsf {b}_x|^2=\frac {|\nabla \mathsf {b}_x^2|^2}{4}$, the Bochner inequality allows us to estimate the left-hand-side above as follows;

(3.84)\begin{align} -\int_{X\setminus \{x\}}\langle \nabla \psi, \nabla (\mathsf{b}_x^2|\nabla \mathsf{b}_x|^2)\rangle \mathrm{d} \mathfrak{m} & ={-}\frac{1}{4}\int_{X\setminus \{x\}}\langle \nabla \psi, \nabla |\nabla \mathsf{b}_x^2|^2\rangle \mathrm{d} \mathfrak{m} \nonumber\\ & \geqslant\frac{1}{2}\int_{X\setminus \{x\}}\psi \left(\frac{(\Delta \mathsf{b}_x^2)^2}{N}+\langle \nabla \Delta \mathsf{b}_x^2, \nabla \mathsf{b}_x^2\rangle \right)\mathrm{d} \mathfrak{m} \nonumber\\ & =\int_{X\setminus \{x\}}\psi \left(2N|\nabla \mathsf{b}_x|^4+N\langle \nabla |\nabla \mathsf{b}_x|^2, \nabla \mathsf{b}_x^2\rangle \right)\mathrm{d} \mathfrak{m}, \end{align}

where we used (3.20) again. Therefore it follows from (3.83) and (3.84) that

(3.85)\begin{align} \int_{X\setminus \{x\}}(2-N)\psi \langle \nabla \mathsf{b}_x^2, \nabla |\nabla \mathsf{b}_x|^2\rangle \mathrm{d} \mathfrak{m} -\int_{X\setminus \{x\}}\langle \nabla (\psi \mathsf{b}_x^2), \nabla |\nabla \mathsf{b}_x|^2\rangle \mathrm{d} \mathfrak{m} \geqslant 0. \end{align}

Let us prove that this inequality (3.85) implies the conclusions. Actually as done in (3.83) and (3.84), it follows from Leibniz’ rule thatFootnote ³

(3.87)\begin{align} & \quad -\int_{X\setminus \{x\}}\langle\nabla \psi, \nabla |\nabla \mathsf{b}_x|^2\rangle\mathrm{d} \mathfrak{m}_{G_x} \nonumber\\ & =\int_{X\setminus \{x\}}\left(-\langle\nabla(\varphi \mathsf{b}_x^2), \nabla |\nabla \mathsf{b}_x|^2\rangle+(2-N)\varphi\langle\nabla \mathsf{b}_x^2,\nabla |\nabla \mathsf{b}_x|^2\rangle\right)\mathrm{d} \mathfrak{m} \geqslant 0 \end{align}

holds, where $\varphi =\mathsf {b}_x^{-2}G_x^2\psi$, which proves the $\mathscr L$-subharmonicity of $|\nabla \mathsf {b}_x|^2$ on $X\setminus \{x\}$.

Similarly we haveFootnote ⁴

(3.89)\begin{align} & \quad -\int_{X\setminus \{x\}}\left\langle \nabla \left(|\nabla \mathsf{b}_x|^2 G_x\right), \nabla \psi \right\rangle \mathrm{d} \mathfrak{m} \nonumber\\ & =\int_{X\setminus \{x\}}\left(-\langle\nabla(\varphi \mathsf{b}_x^2), \nabla |\nabla \mathsf{b}_x|^2\rangle+(2-N)\varphi\langle\nabla \mathsf{b}_x^2,\nabla |\nabla \mathsf{b}_x|^2\rangle\right)\mathrm{d} \mathfrak{m} \geqslant 0, \end{align}

where $\varphi =\psi \mathsf {b}_x^{-N}$, which proves the subharmonicity of $|\nabla \mathsf {b}_x|^2G_x$ on $X \setminus \{x\}$.

Proof. The proof is divided into several steps as follows.

Step 1. Let us prove that $|\nabla \mathsf {b}_x|\in L^\infty (X,\,\mathfrak {m})$. By calculus rules

(3.92)\begin{equation} |\nabla \mathsf{b}_x|=\frac{1}{N-2}G_x^{\frac{N-1}{2-N}}|\nabla G_x|. \end{equation}

Let

(3.93)\begin{equation} C_{\mathrm{opt}, x}:=\lim_{r\rightarrow 0}\ \mathop{\mathrm{ess\text{ }sup}}\limits_{y\in B_r(x)}\ |\nabla \mathsf{b}_x|(y) \left(= \lim_{r\rightarrow 0}\sup_{y\in B_r(x)}\ |\nabla \mathsf{b}_x|(y)\right). \end{equation}

Plugging in (3.51), (3.17) and (3.7) we see that

(3.94)\begin{equation} C_{\mathrm{opt}, x}\leqslant C(N) \nu_x^\frac{1}{N-2}. \end{equation}

Thus we can take $r$ sufficiently small such that

(3.95)\begin{equation} |\nabla \mathsf{b}_x|(y)\leqslant C(N)\nu_x^\frac{1}{N-2}, \quad \mathfrak{m}\text{-a.e.}\ y\in B_r(x)\setminus\{x\}. \end{equation}

On the other hand, combining (3.92) with theorem 3.4 yields that for $\mathfrak {m}$-a.e. $y\in X\setminus \{x\}$, we have

(3.96)\begin{equation} |\nabla \mathsf{b}_x|(y)\leqslant C(N)F_x(\mathsf{d}(x,y))^\frac{N-1}{2-N}H_x(\mathsf{d}(x,y)). \end{equation}

Note that it holds that $sH_x(s)\leqslant F_x(s)$ for any $s\in [1,\,\infty )$. Thus, choosing $R$ sufficiently large, then for any $y\in X\setminus B_R(x)$, we have

(3.97)\begin{equation} |\nabla \mathsf{b}_x|(y)\leqslant C(N)\frac{H_x(\mathsf{d}(x,y))}{F_x(\mathsf{d}(x,y))^{\frac{N-1}{N-2}}}\leqslant C(N)V_X+1 \end{equation}

because of (3.8). Moreover it follows from the continuity of right-hand-side of (3.96) that $|\nabla \mathsf {b}_x|$ is bounded $\mathfrak {m}$-a.e. in $\overline {B_R(x)}\setminus B_r(x)$, which is a compact subset. Therefore $|\nabla \mathsf {b}_x|\in L^\infty (X,\,\mathfrak {m})$.

Step 2. Let us prove that $|\nabla \mathsf {b}_x|(y) \le C_{\mathrm {opt},\, x}$ for any $y \in X \setminus \{x\}$.

Fix $0< r< R$ and choose arbitrarily $\varepsilon >0$. By corollary 3.5, we can find $R_0>R$ sufficiently large such that

(3.98)\begin{equation} G_x(y)< \varepsilon, \quad \forall\ y\in X\setminus B_{R_0}(x) \end{equation}

Take $0< r_0< r$ with

(3.99)\begin{equation} |\nabla \mathsf{b}_x|^2(y)< C_{\mathrm{opt}, x}^2+\varepsilon, \quad \forall\ y\in \overline{B}_{r_0}(x) \setminus \{x\}. \end{equation}

Let $L:=\big \||\nabla \mathsf {b}_x|\big \|_{L^\infty (X,\mathfrak {m})}$ and set

(3.100)\begin{equation} u_x:=|\nabla \mathsf{b}_x|^2G_x-(C_{\mathrm{opt}, x}^2+\varepsilon)G_x-L^2\varepsilon, \end{equation}

which is upper semicontinuous and subharmonic on $X\setminus \{x\}$ because of proposition 3.21.

Let $\Omega :=B_{R_0}(x)\setminus \overline {B_{r_0}(x)}$. Applying the weak maximum principle for upper semicontinuous subharmonic functions [Reference Gigli and Violo31, Proposition 1.15], we see that

(3.101)\begin{equation} \sup_{\Omega}u_x=\sup_{\partial\Omega}u_x\leqslant 0 \end{equation}

which proves

(3.102)\begin{equation} |\nabla \mathsf{b}_x|^2(y)\leqslant C_{\mathrm{opt}, x}^2+\frac{L^2\varepsilon}{G_x(y)}+\varepsilon, \quad \forall\ y \in \Omega. \end{equation}

This observation allows us to conclude that $|\nabla \mathsf {b}_x| \leqslant C_{\mathrm {opt},\, x}$ for $\mathfrak {m}$-a.e. after letting $\varepsilon \to 0^+$ under fixing $r,\, R$. Thus by theorem 3.15 we know that $|\nabla \mathsf {b}_x|(y) \leqslant C_{\mathrm {opt},\, x}$ for any $y \in X \setminus \{x\}$.

Step 3. We claim that $C_{\mathrm {opt},\, x}=\mathscr C_N\nu _x^{\frac {1}{N-2}}$, where this completes the proof. Thanks to (2.30), proposition 3.8 and theorem 3.12, we know that for any $0<\delta <1$

(3.103)\begin{equation} {\unicode{x2A0D}}_{B_r(x)\setminus B_{\delta r}(x)}\left| |\nabla \mathsf{b}_x|^2-\mathscr C_N^2\nu_x^{\frac{2}{N-2}}\right|\mathrm{d} \mathfrak{m} \to 0, \quad \text{as }r \to 0^+. \end{equation}

Take $y_i \to x$ satisfying $|\nabla \mathsf {b}_x|(y_i) \to C_{\mathrm {opt},\, x}$ and let $r_i:=\mathsf {d}(x,\, y_i)$ and consider rescaled spaces;

(3.104)\begin{equation} (X_i, \mathsf{d}_i, \mathfrak{m}_i):=\left(X, \frac{1}{r_i}\mathsf{d}, \frac{1}{r_i^N}\mathfrak{m} \right). \end{equation}

Applying the weak Harnack inequality for $\mathscr L$-superharmonic functions, proposition 6.5, to a lower semicontinuous $\mathscr L$-superharmonic function $C_{\mathrm {opt},\, x}^2-|\nabla \mathsf {b}_{x}^{X_i}|^2 \geqslant 0$, we have

(3.105)\begin{align} \left({\unicode{x2A0D}}_{B_{\frac{1}{4}}^{\mathsf{d}_i}(y_i)}\left|C_{\mathrm{opt}, x}^2-|\nabla \mathsf{b}_{x}^{X_i}|^2\right|^p\mathrm{d} \mathfrak{m}_i \right)^{1/p}& \leqslant C(N) \left(C_{\mathrm{opt}, x}^2-|\nabla \mathsf{b}_x^{X_i}|^2(y_i)\right) \nonumber\\ & =C(N) \left(C_{\mathrm{opt}, x}^2-|\nabla \mathsf{b}_x|^2(y_i)\right) \to 0, \end{align}

where $p=p(N)>0$. Thus recalling $|\nabla \mathsf {b}_x| \in L^{\infty }(X\setminus \{x\},\, \mathfrak {m})$, it holds that

(3.106)\begin{equation} {\unicode{x2A0D}}_{B_{\frac{1}{4}}^{\mathsf{d}_i}(y_i)}\left|C_{\mathrm{opt}, x}^2-|\nabla \mathsf{b}_{x}^{X_i}|^2\right|\mathrm{d} \mathfrak{m}_i \to 0, \end{equation}

namely

(3.107)\begin{equation} {\unicode{x2A0D}}_{B_{\frac{r_i}{4}}(y_i)}\left| C_{\mathrm{opt}, x}^2-|\nabla \mathsf{b}_{x}|^2\right|\mathrm{d} \mathfrak{m} \to 0. \end{equation}

On the other hand, (3.103) implies

(3.108)\begin{equation} {\unicode{x2A0D}}_{B_{\frac{r_i}{4}}(y_i)}\left| \mathscr C_N^2\nu_x^{\frac{2}{N-2}}-|\nabla \mathsf{b}_{x}|^2\right|\mathrm{d} \mathfrak{m} \to 0. \end{equation}

Thus by (3.107) and (3.108) we have $C_{\mathrm {opt},\, x}=\mathscr C_N\nu _x^{\frac {1}{N-2}}$.

Proof. Note

(3.110)\begin{align} & \frac{1}{\mathfrak{m}_i (B_1(x_i))}\int_{B_{\varepsilon}(x_i)}|\nabla \mathsf{b}_{x_i}|^p\mathrm{d} \mathfrak{m}_i \leqslant \frac{\mathfrak{m}_i(B_{\varepsilon}(x_i))}{\mathfrak{m}_i(B_1(x_i))}\mathscr C_N^p \cdot \sup_j \nu_{x_j}^{\frac{p}{N-2}}\nonumber\\ & \quad \to \frac{\mathfrak{m} (B_{\varepsilon}(x))}{\mathfrak{m} (B_1(x))}\mathscr C_N^p \cdot \sup_j\nu_{x_j}^{\frac{p}{N-2}}. \end{align}

Since the right-hand-side of (3.110) is small if $\varepsilon$ is small, combining this with theorem 3.12 completes the proof.

Proof. Thanks to (3.16) and (3.6), putting $\mathsf {b}_x(x):=0$ gives a unique continuous extention of $\mathsf {b}_x$ on $X$. Since $\{x\}$ is null with respect to the $2$-Sobolev capacity because of the finiteness of $\nu _x$ (see [Reference Björn and Björn9]), we know $\mathsf {b}_x \in W^{1, 2}_{\mathrm {loc}}(X,\, \mathsf {d},\, \mathfrak {m})$. Then the first statement comes from arguments in the last step in the proof of theorem 3.22 with the (local) Sobolev-to-Lipschitz property and corollary 3.23. Moreover corollary 3.23 with proposition 3.8 allows us to obtain (3.112).

Proof. Since (3.51), (3.92) and corollary 3.24 yield (under a suitable limit $y \to x$)

(3.114)\begin{align} \frac{|\nabla G_x|(y)}{\mathsf{d} (x,y)^{1-N}}& =(N-2)\left(\mathsf{d} (x,y)G_x(y)^{\frac{1}{N-2}}\right)^{N-1}|\nabla \mathsf{b}_x|(y) \nonumber\\ & \to (N-2) \left( \frac{1}{\mathscr C_N\nu_x^{\frac{1}{N-2}}}\right)^{N-1}\mathscr C_N\nu_x^{\frac{1}{N-2}} \nonumber\\ & =\frac{1}{N\nu_x} \end{align}

we conclude.

Proof. Firstly we discuss the case when $V_X=0$. Then we can follow the same arguments as in the proof of [Reference Colding16, Theorem 2.12]. Namely we can estimate as $\mathsf {d}(x,\, y) \to \infty$,

(3.119)\begin{align} |\nabla \mathsf{b}_x|(y) & =\frac{G_x^{\frac{N-1}{2-N}}}{N-2}|\nabla G_x|(y) \nonumber\\ & \leqslant \frac{G_x(y)^{\frac{N-1}{2-N}}}{N-2}\cdot C(N) \cdot G_x(y) \cdot \frac{1}{\mathsf{d}(x,y)} \nonumber\\ & \leqslant C(N)\left( \frac{F_x(\mathsf{d}(x, y))}{\mathsf{d}(x,y)^{2-N}}\right)^{\frac{1}{2-N}} \to 0, \end{align}

where we used the gradient estimates on positive harmonic functions obtained in [Reference Jiang42, Theorem 1.2] in the first inequality above and we also used (3.8) and (3.16) in the last inequality and in the limit. Thus we obtain the conclusion in this case.

Next we consider the case when $V_X>0$. The first statement, (3.115), is a direct consequence of corollary 3.23 and (2) of corollary 2.9. The remaining one (3.117) (or (3.118)) follows from an argument similar to the proof of corollary 3.25.

Proof. For the first statement, we just follow the proof of [Reference Fogagnolo, Mazzieri and Pinamonti26, Theorem 3.5]. Namely our goal is to prove for any $t>0$

(3.126)\begin{equation} |\nabla \mathsf{b}_x| \leqslant \sup_{\{\mathsf{b}_x=t\}}|\nabla \mathsf{b}_x|,\quad \text{on }\{\mathsf{b}_x \ge t\} \end{equation}

because then for any $s \le t$ we have

(3.127)\begin{equation} f_x(s)= \sup_{\{\mathsf{b}_x \ge s\}}|\nabla \mathsf{b}_x| \geqslant \sup_{\{\mathsf{b}_x \ge t\}}|\nabla \mathsf{b}_x| = f_x(t) \end{equation}

which completes the proof of the first statement. Thus let us focus on the proof of (3.126). Note that $G_x$ is $\mathcal {L}$-harmonic on $X \setminus \{x\}$. Fixing a sufficiently large $T>t$, consider an $\mathcal {L}$-subharmonic function $\varphi$ on $X \setminus \{x\}$ defined by

(3.128)\begin{equation} \varphi(y):=|\nabla \mathsf{b}_x|^2(y)-f_x(t)-\frac{\mathscr C_N^2\nu_x^{\frac{2}{N-2}}}{T^{N-2}G_x(y)}. \end{equation}

It is trivial that $\varphi \le 0$ holds on $\{ \mathsf {b}_x =t \} \cup \{\mathsf {b}_x= T\}$. Thus the strong maximum principle, proposition 6.7, shows $\varphi \le 0$ on $\{t \leqslant \mathsf {b}_x \leqslant T\}$. Then letting $T \to \infty$ proves (3.126).

On the other hand (3.124) is a direct consequence of theorem 3.22 and corollary 3.24. Thus let us focus on the proof of (3.125). The proof is divided into the following $2$ cases.

Case 1: $V_X=0$.

By the same argument as in (3.97), we see that for any $\epsilon >0$, there exists $R>1$ such that

(3.129)\begin{equation} |\nabla \mathsf{b}_x|(y) \leqslant C(N)V_X+\epsilon=\epsilon \end{equation}

holds for any $y \in X \setminus B_R(x)$, namely we have $\limsup _{\mathsf {d}_x(y) \to \infty }|\nabla \mathsf {b}_x|(y) \leqslant C(N) V_X=0$ which proves (3.124) in this case.

Case 2: $V_X>0$.

In this case the proof is similar to Step 3 of the proof of theorem 3.22 with (3.126). Let $C_{\mathrm {opt},\, \infty }:=\lim _{t \to \infty }f_x(t)$. Take $x_t \in \{\mathsf {b}_x=t\}$ with $|\nabla \mathsf {b}_x|(x_t)=f_x(t)$, and let $0\epsilon _t:=f_x(t)^2-C_{\mathrm {opt},\, \infty }^2\geqslant 0$. Note that $C_{\mathrm {opt},\, \infty }^2-|\nabla \mathsf {b}_x|^2 +\varepsilon _t \geqslant 0$ holds on $\{\mathsf {b}_x\geqslant t\}$.

On the other hand, thanks to (3.52), for any sufficiently large $t>1$, we have $B_{\delta t}(x_t) \subset \{\mathsf {b}_x\geqslant \frac {t}{2}\}$, where $\delta =\delta (N,\, V_X)$ is a positive constant depending only on $N,\, V_X$. In particular

(3.130)\begin{equation} C_{\mathrm{opt}, \infty}^2-|\nabla \mathsf{b}_x|^2 +\varepsilon_{\frac{t}{2}} \geqslant 0,\quad \text{on }B_{\delta t}(x_t). \end{equation}

Then, after a rescaling $t^{-1}\mathsf {d},\, \mathfrak {m} (B_t(x))^{-1}\mathfrak {m}$, applying the weak Harnack inequality for $\mathcal {L}$-superharmonic function, proposition 6.5, to $C_{\mathrm {opt},\, \infty }^2-|\nabla \mathsf {b}_x|^2 +\varepsilon _{\frac {t}{2}} \geqslant 0$, we have as $t \to \infty$

(3.131)\begin{equation} \left({\unicode{x2A0D}}_{B_{\frac{t}{4}}(x_{t})}\left| C_{\mathrm{opt}, \infty}^2-|\nabla \mathsf{b}_{x}|^2+\varepsilon_{\frac{t}{2}}\right|^p\mathrm{d} \mathfrak{m}\right)^{1/p} \leqslant C(N) \left(C_{\mathrm{opt}, \infty}^2-|\nabla \mathsf{b}_{x}|^2(x_t)+\varepsilon_{\frac{t}{2}}\right) \to 0, \end{equation}

for some $p=p(N)>0$, namely

(3.132)\begin{equation} {\unicode{x2A0D}}_{B_{\frac{t}{4}}(x_{t})}\left| C_{\mathrm{opt}, \infty}^2-|\nabla \mathsf{b}_{x}|^2\right|\mathrm{d} \mathfrak{m} \to 0. \end{equation}

Combining this with (3.116) yields $C_{\mathrm {opt},\, \infty }=\mathscr C_NV_X^{\frac {1}{N-2}}$ which completes the proof of (3.125).

Proof. First of all, let us prove that $X \setminus \{x\}$ is connected. Let $U$ be a connected component of $X\setminus \{x\}$. As mentioned in the beginning of the proof of corollary 3.24, since $\{x\}$ has a null $2$-capacity, the indicator function $\chi _U$ of $U$ is in $H^{1,2}_{\mathrm {loc}}(X,\, \mathsf {d},\, \mathfrak {m})$ with $|\nabla \chi _U|=0$. Thus the (local) Sobolev-to-Lipschitz property shows $\chi _U= 1$ for $\mathfrak {m}$-a.e., which implies $X\setminus \{x\}=U$. Thus $X\setminus \{x\}$ is connected.

Then applying the strong maximum principle, proposition 6.7, for $|\nabla \mathsf {b}_x|^2$ yields $|\nabla \mathsf {b}_x|\equiv \mathscr C_N\nu _x^\frac {1}{N-2}$ in $X \setminus \{x\}$. Letting $u:={\mathsf {b}_x^2}/ (2\mathscr C_N^2\nu _x^\frac {2}{N-2})$, we have $|\nabla \sqrt {2u}|^2=1$ and thus, by (3.20), $\Delta u=N$ for $\mathfrak {m}$-a.e. These observations allow us to apply [Reference Gigli and Violo31, Theorem 5.1] (see also Theorem 2.8) to get the conclusion.

Proof. Let us prove (1). Theorem 4.1 yields that $(X,\, \mathsf {d},\, \mathfrak {m})$ is isomorphic to an $N$-metric measure cone over an $\operatorname {RCD}(N-2,\, N-1)$ space. In particular it must be isomorphic to a tangent cone at $x$. Thus we conclude.

Next let us prove (2). The sharp gradient estimate, theorem 3.22, yields

(4.2)\begin{equation} \lim_{r \to 0^+}\frac{\mathscr{H}^N(B_r(x))}{\omega_Nr^N} \ge 1, \end{equation}

thus $x$ is an $N$-regular point because of [Reference De Philippis and Gigli20, Corollary 1.7] (recall (2.15)). Then the conclusion follows from the first statement (1).

Proof. As in the proof of corollary 3.14, it is enough to consider the case when $p=2$. Then the proof is done by a contradiction, thus assume the conclusion fails. Then there exist sequences of positive numbers $\delta _i\to 0^+$ and of pointed $\operatorname {RCD}(0,\,N)$ spaces $(X_i,\,\mathsf {d}_i,\,\mathfrak {m}_i,\,x_i)$ such that $\nu _{x_i} \leqslant \nu$, $\frac {s}{\mathfrak {m}_i(B_s(x_i))} \leqslant \varphi (s)$ for $\mathscr {H}^1$-a.e. $s \in [1,\, \infty )$, that

(4.7)\begin{equation} |\nabla_i \mathsf{b}_{x_i}|(z_i)\geqslant \mathscr C_N\nu_{x_i}^\frac{1}{N-2}-\delta_i,\quad \text{for some }z_i \in \overline{B}_R(x_i) \setminus B_r(x_i) \end{equation}

and that for any $\operatorname {RCD}(N-2,\,N-1)$ space $(Y,\,\mathsf {d}_Y,\,\mathfrak {m}_Y)$, we have

(4.8)\begin{equation} \inf_i \left(\mathsf{d}_{\mathrm{pmGH}}\left((X_i,\mathsf{d}_i,\mathfrak{m}_i,x_i),(C(Y),\mathsf{d}_{C(Y)},\mathfrak{m}_{C(Y)},O_Y)\right)+|\nu_{x_i}-\nu_{O_Y}|\right)>0 \end{equation}

or

(4.9)\begin{equation} \inf_i\left(\left\| \mathsf{b}_{x_i}-\mathscr C_N\nu_{x_i}^{\frac{1}{N-2}}\mathsf{d}_{x_i}\right\|_{L^{\infty}(B_R(x_i))}\!+\!\left({\unicode{x2A0D}}_{B_R(x_i)}\left| \nabla \left(\mathsf{b}_{x_i} \!-\!\mathscr C_N\nu_{x_i}^{\frac{1}{N-2}}\mathsf{d}_{x_i}\right)\right|^p\mathrm{d} \mathfrak{m}\right)^{1/p} \right)\!>\!0. \end{equation}

Theorem 3.11 allows us to conclude that after passing to a subsequence, $(X_i,\,\mathsf {d}_i,\,\mathfrak {m}_i,\,x_i)$ pmGH converge to a pointed non-parabolic $\operatorname {RCD}(0,\, N)$ space $(X,\,\mathsf {d},\,\mathfrak {m},\,x)$, and $\nu _{x_i} \to \mu$ for some $\mu \in [0,\, \infty )$. On the other hand, the lower semicontinuity of $N$-volume densities (3.34) implies

(4.10)\begin{equation} \nu_{x}\leqslant \lim_{i\to \infty}\nu_{x_i}=\mu\leqslant \nu<\infty. \end{equation}

Consider $\mathscr L$-superharmonic lower semicontinuous functions on $X_i\setminus \{x_i\}$;

(4.11)\begin{equation} u_i:=\mathscr C_N^2\nu_{x_i}^\frac 2{N-2}-|\nabla_i \mathsf{b}_{x_i}|^2 \geqslant 0. \end{equation}

Fix $0< s<\frac {r}{4}$. Applying the weak Harnack inequality, proposition 6.5, to the nonegatively valued $\mathscr L$-superharmonic function $u_i$, we have

(4.12)\begin{equation} \left({\unicode{x2A0D}}_{B_{2s}(z_i)}u_i^p\, \mathrm{d}\mathfrak{m}_i\right)^\frac 1 p\leqslant C\inf_{B_s(z_i)} u_i\leqslant C u_i(z_i)\leqslant C\delta_i. \end{equation}

where $p=p(N,\, s)>0$. Thus recalling that $u_i$ is uniformly bounded, we have

(4.13)\begin{equation} {\unicode{x2A0D}}_{B_{2s}(z_i)}\left|\mathscr C_N^2\nu_{x_i}^\frac 2{N-2}-|\nabla_i \mathsf{b}_{x_i}|^2\right|\, \mathrm{d}\mathfrak{m}_i\rightarrow 0. \end{equation}

Note that with no loss of generality we can assume that $z_i$ converge to a point $z \in \overline {B}_R(x)\setminus B_r(x)$. Moreover the dominated convergence theorem yields the convergence of $\frac {s}{\mathfrak {m}_i(B_s(x_i))}$ to $\frac {s}{\mathfrak {m} (B_s(x))}$ in $L^1([1,\, \infty ),\, \mathscr {H}^1)$. Thus theorem 3.12 shows that $\mathsf {b}_{x_i}$ $W^{1,2}$-strongly converge to $\mathsf {b}_{x}$ on $B_{2s}(z)$. Therefore (4.13) with (4.10) implies

(4.14)\begin{equation} {\unicode{x2A0D}}_{B_{2s}(z)}|\nabla \mathsf{b}_x|^2\mathrm{d} \mathfrak{m} = \mathscr C_N^2\mu^{\frac{2}{N-2}} \geqslant \mathscr C_N^2\nu_x^{\frac{2}{N-2}}. \end{equation}

In particular theorem 3.22 shows $\mu =\lim _{i\to \infty }\nu _{x_i}=\nu _x$ and

(4.15)\begin{equation} |\nabla b_x|=\mathscr C_N\nu_x^\frac 1 {N-2}, \quad \text{for }\mathfrak{m}\text{-a.e. in }B_{2s}(z). \end{equation}

Thus, theorem 4.1 allows us to conclude that the limit space is isomorphic to the $N$-metric measure cone over an $\operatorname {RCD}(N-2,\, N-1)$ space, which contradicts (4.8) and (4.9).

Proof. The first statement is a direct consequence of (the proof of) theorem 4.3 with the Bishop inequality $\nu _x \leqslant \omega _N$.

Proof. Let $r_i:=\mathsf {d}(x,\, y_i)$ and consider a rescaled pointed non-parabolic non-collapsed $\operatorname {RCD}(0,\, N)$ space;

(4.22)\begin{equation} (X_i, \mathsf{d}_i, \mathscr{H}^N_{\mathsf{d}_i}, x_i):=\left( X, \frac{1}{r_i}\mathsf{d}, \frac{1}{r_i^N}\mathscr{H}^N_{\mathsf{d}}, x\right). \end{equation}

Recalling (3.30) we have for any $r \geqslant 1$

(4.23)\begin{equation} F_{x_i}^{X_i}(r)=\frac{F_x^X\left( rr_i\right)}{r_i^{2-N}} \leqslant \frac{r^{2-N}}{(N-1)\tau}, \end{equation}

where we used (3.5) in the final inequality. Thus corollary 4.4 allows us to conclude that $(X_i,\, \mathsf {d}_i,\, \mathscr {H}^N_{\mathsf {d}_i},\, x_i)$ is pmGH close to the $N$-dimensional Euclidean space. In particular a tangent cone at infinity is also pmGH close to the $N$-dimensional Euclidean space. Therefore the volume convergence result, [Reference De Philippis and Gigli20, Theorem 1.3], implies (4.21). Thus we get the first statement. The remaining statements come from this with the same arguments as in [Reference Cheeger and Colding14, Theorems A.1.11].