To begin we fix some sequence $k_{\nu}\rightarrow\infty$ so that the scalings of the based space $(X_{\infty},p)$ by $\sqrt{k_{\nu}}$ converge to a tangent cone $C(Y)$. For a while we focus attention on this cone. Write $|z|$ for the distance from the vertex. Let $\Sigma_{Y}\subset Y$ be the singular set and $Y^{{\rm reg}}=Y\setminus\Sigma$.

The only information about the singular set which we need is contained in the following proposition. This is very likely a standard fact but the proof is quite short so we include it.

Recall that $Y$ has dimension $2n-1$. For clarity in this proof we write $N=2n-1$. A simple argument using the noncollapsing condition (1.2) and the Bishop inequality in the original manifolds shows that there are fixed numbers $\overline{c},\underline{c}>0$ such that for $r\leq 1$ and any metric ball $B_{r}$ in $Y$ we have

(3.1)

$$\underline{c}r^{N}\leq\text{Vol}(B_{r})\leq\overline{c}r^{N}.$$

We know that $\Sigma_{Y}$ is a compact set of Hausdorff dimension strictly less than $N-2$. By the definition of Hausdorff dimension we can find a number $\lambda\in(0,N-2)$ with the following property. For any $\epsilon>0$ there is a cover of $\Sigma_{Y}$ by a finite number of balls $B_{r_{i}/2}(p_{i})$ such that

(3.2)

$$\sum r_{i}^{N-2-\lambda}<\epsilon.$$

We write $B_{i}=B_{r_{i}}(p_{i})$ so, in an obvious notation, the cover is by the balls $\frac{1}{2}B_{i}$. By the Vitali argument we can suppose that the balls $\frac{1}{10}B_{i}$ are disjoint. We take $\epsilon<1$ so for each $i$ we obviously have $r_{i}\leq\epsilon^{1/(N-2-\lambda)}<1$.

Let $\phi(t)$ be a standard cut-off function, vanishing for $t\geq 2$, equal to $1$ when $t\leq 1$ and with derivative bounded by $2$. Define

$$f_{i}(y)=\phi(r_{i}^{-1}d(y,p_{i})).$$

Thus $f_{i}$ is supported in $B_{i}$ and equal to $1$ in $\frac{1}{2}B_{i}$. This function need not be smooth but it is Lipschitz and differentiable almost everywhere, with $|\nabla f_{i}|\leq 2r_{i}^{-1}$. Set $f=\sum f_{i}$. Let $\Psi(t)$ be a cut-off function, equal to $1$ when $t\geq 9/10$, with $\Psi(0)=0$ and with derivative bounded by $2$. Put $g_{0}=\Psi\circ f$. Then $g_{0}$ is equal to $1$ on a neighbourhood of $\Sigma_{Y}$ and is supported in the $2\epsilon^{1/N-2-\lambda}$-neighbourhood of $\Sigma_{Y}$. Also we have

$$\|\nabla g_{0}\|_{L^{2}}\leq 2\|\nabla f\|_{L^{2}}.$$

We claim that $\|\nabla f\|^{2}_{L^{2}}\leq C_{5}\epsilon$ for some fixed $C_{5}$, depending only on $\underline{c},\overline{c}$. Given this claim we can make $\|\nabla g_{0}\|_{L^{2}}$ as small as we please and then finally approximate $g_{0}$ by a smooth function $g$ to achieve our result. (Note that this approximation only involves working over a compact subset of $Y^{{\rm reg}}$. )

To establish the claim, divide the index set into subsets

$$I_{\alpha}=\{i:2^{-\alpha-1}\leq r_{i}<2^{-\alpha}\},$$

for $\alpha\geq 0$. A simple packing argument, using the fact that the balls $\frac{1}{10}B_{i}$ are disjoint, shows that there is a fixed number $C_{6}$ with the following property. If $j\in I_{\alpha}$ then for each fixed $\beta\leq\alpha$ there are at most $C_{6}$ balls $B_{i}$ with $i\in I_{\beta}$ which intersect $B_{j}$. Now we have

$$\|\nabla f\|^{2}_{L^{2}}\leq\sum_{i,j}\int|\nabla f_{i}|\ \ |\nabla f_{j}|.$$

Thus

$$\|\nabla f\|^{2}_{L^{2}}\leq 2\sum_{i,j:r_{j}\leq r_{i}}\int|\nabla f_{i}|\ |\nabla f_{j}|.$$

For fixed $j$ there are at most $C_{6}(1+\log_{2}(r_{j}^{-1}))$ terms which contribute to this last sum. For each term $|\nabla f_{i}|\leq 2r_{i}^{-1}\leq 2r_{j}^{-1}$ and the integrand is supported on the ball $B_{j}$ of radius $2r_{j}$. So for fixed $j$ the contribution to the sum is bounded by

$$8C_{6}(1+\log_{2}(r_{j}^{-1}))r_{j}^{-2}\overline{c}(2r_{j})^{N}.$$

Hence, summing over $j$,

$$\|\nabla f\|^{2}_{L^{2}}\leq 2^{N+3}C_{6}\overline{c}\sum r_{j}^{N-2}(\log_{2}(r_{j}^{-1})+1).$$

We can find a number $C_{7}$ such that for $t\geq 1$ we have $1+\log t\leq C_{7}t^{\lambda}$. Thus

$$\|\nabla f\|^{2}_{L^{2}}\leq 2^{N+3}C_{6}C_{7}\overline{c}\sum r_{j}^{N-2-\lambda}\leq 2^{N+3}C_{6}C_{7}\overline{c}\epsilon.$$

We pick some base point $y_{0}$ in $Y^{{\rm reg}}$. We will need 4 parameters $\rho,\epsilon,\delta,R$ in our basic construction, where $\rho,\epsilon,\delta$ will be “small” and $R$ “large”. In particular $\delta<<\rho<<1<<R$.

First we fix $\rho$ so that $\exp(-\rho^{2}/4)\geq 3/4$ and $\rho\leq(16K_{1})^{-1}$ where $K_{1}$ is the constant in our first derivative estimate. We take $u_{*}=\rho y_{0}\in C(Y)$, with the obvious notation. Fix any neighbourhood $D$ of $u_{*}$ whose closure does not meet the singular set in $C(Y)$. For any $\epsilon$ let $Y_{\epsilon}$ be the set of points of distance greater than $\epsilon$ from $\Sigma$. Let $U_{\epsilon,\delta,R}$ be the set of points $z$ in $C(Y_{\epsilon})$ such that $\delta<|z|<R$. We choose the parameters so that $U_{\epsilon,\delta,R}$ contains the closure of $D$. We consider a smooth compactly supported cut-off function $\beta$ on $U_{\epsilon,\delta,R}$. For such a function we set

$$E_{\beta}=\int e^{-|z|^{2}/2}|\nabla\beta|^{2}.$$

To see this we take $\beta=\beta_{\delta}\beta_{R}\beta_{\epsilon}$ where

• •

  $\beta_{\delta}$ is a standard cut-off function of $|z|$ equal to $1$ for $|z|>2\delta$.

• •

  $\beta_{R}$ is likewise a standard cut-off function of $|z|$, equal to $1$ for $|z|<R/2$.

• •

  $\beta_{\epsilon}=1-(g\circ\varpi)$ where $g$ is a function on $C(Y)$ of the kind constructed in Proposition (3.5) and $\varpi$ is the radial projection from the cone minus the vertex to $Y$.

Then the lemma follows from elementary calculations.