6.4. Geometric singularity and normalized limit measure [055D]

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6.4. Geometric singularity and normalized limit measure

The goal of this subsection is to understand the measured Gromov-Hausdorff limits of the sequence of incomplete Calabi-Yau metrics $(\mathcal{M}_{T},\omega_{T,CY})$ (scaled to fixed diameter) constructed in Theorem 6.3. As can be easily seen, the results are parallel to the statements in Theorem 1.1, and in Section 7 we shall not reproduce the arguments from here.

To begin with, we recall the notion of measured Gromov-Hausdorff convergence. We refer the readers to [CC97] for the general theory about this.

Definition 6.11 (Measured Gromov-Hausdorff convergence).

Let $(M_{j}^{m},g_{j},p_{j})$ be a sequence of Riemannian manifolds with $\Ric_{g_{j}}\geq-(m-1)$ such that

(6.136)

(M_{j}^{m},g_{j},p_{j})\xrightarrow{GH}(X_{\infty},d_{\infty},p_{\infty})

for some metric space $(X_{\infty},d_{\infty},p_{\infty})$ , then by passing to a subsequence, the renormalized measures

(6.137)

d\underline{\nu}_{j}\equiv\frac{\dvol_{g_{j}}}{\Vol_{g_{j}}(B_{1}(p_{j}))}

converge to a Radon measure $d\underline{\nu}_{\infty}$ on $X_{\infty}$ which is called the renormalized limit measure. The Gromov-Hausdorff convergence together with the convergence of the renormalzied measures is called the measured Gromov-Hausdorff convergence.

In the general context of collapsed sequences with Ricci curvature bounded from below, $d\underline{\nu}_{\infty}$ behaves quite differently from the Hausdorff measures on $X_{\infty}$ induced by the limiting metric $d_{\infty}$ . In our specific context, $d\underline{\nu}_{\infty}$ has an explicit form and it effectively reveals the geometric singularity information in the collapsing spaces.

Now return to our context. We are interesting in the measured Gromov-Hausdorff limits of $(\mathcal{M}_{T},\omega_{T},d\underline{\nu}_{T})$ , where $d\underline{\nu}_{T}=\frac{1}{n!}\omega_{T}^{n}$ is the volume measure of the metric $\omega_{T}$ . Using the error estimate in Proposition 4.23, the convergence is in fact dominated by large scale geometries of the neck metric $(\mathcal{M}_{T},\omega_{T})$ constructed in Section 4.1. So we shall only perform the calculation using the metrics $\omega_{T}$ , and the latter are fairly explicit by construction.

Gromov-Hausdorff limit:

By construction and direct calculation one sees that $g_{T}$ in large scale is approximated by the $1$ dimensional metric tensor $T^{\frac{(2-n)(n+1)}{n}}L_{T}(z)^{n-1}dz^{2}$ . In particular, the diameter is of order $T^{\frac{n+1}{n}}$ . This suggests rescaling the metric $g_{T}$ by $T^{-\frac{2(n+1)}{n}}$ in order to obtain bounded diameter. Indeed, upon the change of variable $z=T\cdot\xi$ , we see $T^{-\frac{2(n+1)}{n}}g_{T}$ converges to the one dimensional metric $(1+k_{\mp}\xi)^{n-1}d\xi^{2}$ , $\xi\in[-k_{-}^{-1},-k_{+}^{-1}]$ in the Gromov-Hausdorff sense.

The above limit can be transformed into the standard metric on the unit interval $(\mathbb{I},dv^{2})$ via a constant rescaling and the following coordinate change

(6.138)

\begin{cases}1+k_{+}(\frac{1}{k_{-}}-\frac{1}{k_{+}})v=(1+k_{+}\xi)^{\frac{n+1}{2}},\ \ \xi>0;\\ 1+k_{-}(\frac{1}{k_{-}}-\frac{1}{k_{+}})v=(1+k_{-}\xi)^{\frac{n+1}{2}},\ \ \xi<0.\end{cases}

Renormalized limit measure:

Again we first calculate by definition

(6.139)

\text{Vol}(\mathcal{M}_{a\leq z\leq b},\omega_{T})=\int_{a}^{b}dz\frac{T^{2-n}}{(n-1)!}\int_{D}\tilde{\omega}(z)^{n-1}.

Upon the change of variable $z=T\cdot\xi$ , this we get

(6.140)

\text{Vol}(\mathcal{M}_{a\leq\xi\leq b},\omega_{T})=T^{2}\int_{a}^{b}d\xi\int_{D}\frac{1}{(n-1)!}(1+k_{\pm}\xi)^{n-1}\omega_{D}^{n-1}.

So up to constant, the renormalized limit measure has density function given by $(1+\pm\xi)^{n-1}d\xi$ . Changing to the $v$ -variable this becomes (again up to constant multiplication)

(6.141)

d\underline{\nu}_{\infty}=\begin{cases}(\frac{v}{-k_{+}}+\frac{1}{k_{-}-k_{+}})^{\frac{n-1}{n+1}}dv,\ \ v\in[\frac{k_{+}}{k_{-}-k_{+}},0];\\ (\frac{v}{-k_{-}}+\frac{1}{k_{-}-k_{+}})^{\frac{n-1}{n+1}}dv,\ \ v\in[0,\frac{k_{-}}{k_{-}-k_{+}}].\end{cases}

Fibration structure:

There is an obvious fibration of $\mathcal{M}_{T}$ over $[T_{-},T_{+}]$ using the coordinate function $z$ . Composing with above coordinate changes, we obtain a fibration

(6.142)

\mathcal{F}_{T}:\mathcal{M}_{T}\rightarrow\mathbb{I};\bm{x}\mapsto v(\bm{x}).

It is clear that for any $v\neq 0$ , $\mathcal{F}_{T}^{-1}(v)$ is an $S^{1}$ bundle over $D$ , whose first Chern class is given by $c_{1}(L_{\pm})$ depending on the sign of $v$ , and $\mathcal{F}_{T}^{-1}(0)$ is an singular $S^{1}$ fibration over $D$ , with vanishing circles along $H$ .

Bubble classification:

From our analysis in Section 4.3, it is clear that suitable rescalings around the vanishing circles in $\mathcal{F}_{T}^{-1}(0)$ are given by the product space $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ . Also suitable rescalings around the ends $z=T_{\pm}$ gives the incomplete Calabi model spaces.

We close this section by giving the following remarks regarding the regularity of the renormalized limit measure.

Remark 6.11.1.

It can be seen from the above formulae that the limiting density function $\mathscr{V}_{\infty}=\frac{d\underline{\nu}_{\infty}}{dv}$ is a Lipschitz function on $\mathbb{I}$ and it is smooth everywhere in the interior of $\mathbb{I}$ except at $v=0$ . On the other hand, the singular fiber of $\mathcal{F}$ precisely appears at $t=0$ . So in our context, the singularity of the renormalized limit measure $d\underline{\nu}_{\infty}$ effectively characterizes the singularity behavior of the collapsing geometry.

Remark 6.11.2.

By Cheeger-Colding (see [CC00], theorem 4.6), in the regular set $\mathcal{R}$ of a general Ricci-limit space, the density function $\mathscr{V}_{\infty}$ of the renormalized limit measure always exists and is Hölder continuous. Our example tells us that, in general, one cannot expect the regularity of $\mathscr{V}_{\infty}$ to be differentiable in $\mathcal{R}$ (even though $\mathcal{R}$ is a smooth Riemannian manifold). We thank Shouhei Honda for pointing this out.

Remark 6.11.3.

If we use rescale the metrics further around the point $v=0$ such that the sequence of spaces collapse to the complete real line $\mathbb{R}$ , then $d\underline{\nu}_{\infty}$ coincides with the standard Lebesgue measure. In particular, the singularity at $v=0$ disappears. This fact can be quickly seen by scaling-up the coordinates $v$ . This is compatible with the general theory of Ricci-limit spaces. That is, due to Cheeger-Colding, the renormalized limit measure always splits off the Lebesgue measure of $\mathbb{R}$ if the limit space isometrically splits off $\mathbb{R}$ (see proposition 1.35 in [CC97] for more details).