On the small resolutions above, the potential limit for a closed background gives smooth convergence of the metrics away from the exceptional curves. Global distances, however, are also affected by paths through neighborhoods of these curves, so their convergence requires control of the diameters of such neighborhoods. The criterion of proposition 15.19.22 separates the two controls: uniform convergence on compact subsets of the regular part, and small diameters of neighborhoods of the contracted sets, in the limiting metric and uniformly in the approximating metrics. We first recall the Gromov–Hausdorff distance.
Definition 15.19.19.0EB9DF723176 For nonempty subsets $E,F$ of a metric space $(W,d_ W)$, their Hausdorff distance is
\[ d_{\mathrm H}^ W(E,F)=\max \left\{ \sup _{x\in E}\inf _{y\in F}d_ W(x,y), \sup _{y\in F}\inf _{x\in E}d_ W(x,y)\right\} . \]
For nonempty bounded metric spaces $E,F$, put
\[ d_{\mathrm{GH}}(E,F) =\inf _{W,\iota ,\jmath }d_{\mathrm H}^ W(\iota (E),\jmath (F)), \]
where $\iota ,\jmath $ are isometric embeddings into a common metric space $W$. A correspondence is a subset $R\subset E\times F$ whose two coordinate projections are surjective. Its distortion is
\[ \operatorname {dis}R =\sup _{(x,y),(x',y')\in R} |d_ E(x,x')-d_ F(y,y')|. \]
Lemma 15.19.20.022849340F28 For any correspondence $R$ between nonempty bounded metric spaces,
\[ \frac12|\operatorname {diam}E-\operatorname {diam}F| \leq d_{\mathrm{GH}}(E,F) \leq \frac12\operatorname {dis}R. \]
Proof.
Choose $\varepsilon >0$ with $2\varepsilon \geq \operatorname {dis}R$. Give the disjoint union $E\sqcup F$ its original distances within each space, and use the symmetric cross distance
\[ D(x,y)=\inf _{(x_0,y_0)\in R} \bigl (d_ E(x,x_0)+\varepsilon +d_ F(y_0,y)\bigr ), \qquad x\in E,\ y\in F. \]
It is finite and at least $\varepsilon $. The triangle inequalities involving one cross distance follow from the triangle inequality within the corresponding space and the infimum. For two cross distances, take $(x_0,y_0),(x_1,y_1)\in R$. The distortion bound gives
\[ \begin{split} d_ E(x,x') & \leq d_ E(x,x_0)+d_ F(y_0,y_1) +2\varepsilon +d_ E(x_1,x')\\ & \leq d_ E(x,x_0)+d_ F(y_0,y)+\varepsilon +\varepsilon +d_ F(y,y_1)+d_ E(x_1,x'). \end{split} \]
Taking the two infima gives $d_ E(x,x')\leq D(x,y)+D(y,x')$. The same argument with $E,F$ exchanged treats the other case. Hence $D$ is a metric extending both metrics. Each point has a correspondent at cross distance $\varepsilon $, so the Hausdorff distance between $E$ and $F$ in $(E\sqcup F,D)$ is at most $\varepsilon $. Letting $\varepsilon $ decrease to half the distortion proves the upper bound, including zero distortion.
For the lower bound, suppose two isometric images have Hausdorff distance $h$. For every $\delta >0$, two points in the first image have points in the second within $h+\delta $. Their distance is at most $\operatorname {diam}F+2h+2\delta $. Taking the supremum and then $\delta \downarrow 0$ gives $\operatorname {diam}E\leq \operatorname {diam}F+2h$. Exchanging the images and taking the infimum over embeddings proves the lower bound.
$\square$
We now work in a purely metric setting, which contains the nodal one. Let $Y$ be a compact metrizable space and let $S=\{ s_1,\ldots ,s_ N\} \subset Y$ be finite. Assume $Y^\circ =Y\setminus S$ is a dense connected smooth manifold. Let $g_0$ be a smooth Riemannian metric on $Y^\circ $, and denote its intrinsic length distance by $d_0$.
Lemma 15.19.21.E3C9C96E314E Suppose every $s_ a$ has a neighborhood basis of open sets $V_{a,r}$ such that
\[ \lim _{r\downarrow 0}\operatorname {diam}_{d_0}(V_{a,r}\setminus S)=0. \]
Then the metric completion $(\widehat Y,\widehat d_0)$ of $(Y^\circ ,d_0)$ is compact, and the identity on $Y^\circ $ extends to a homeomorphism $\iota :Y\to \widehat Y$.
Proof.
The neighborhoods can be chosen nested, with closures avoiding the other points of $S$. Every sequence of regular points converging to $s_ a$ is $d_0$-Cauchy: its tail lies in each prescribed neighborhood, whose punctured diameter tends to zero. Interleaving two such sequences shows that they define the same completion point, denoted by $\widehat s_ a$. Density of $Y^\circ $ guarantees the existence of these sequences.
For each prescribed positive diameter, remove sufficiently small neighborhoods of all the points of $S$. Their complement is a compact subset of $Y^\circ $ and hence is totally bounded for $d_0$. The removed punctured neighborhoods each have the prescribed small diameter. Thus $Y^\circ $ is totally bounded, and its completion is compact.
The points $\widehat s_ a$ are distinct from one another and from the regular points. To see this, fix $s_ a$ and a different point $y\in Y$. Choose an open neighborhood $V$ of $s_ a$ whose closure avoids $y$ and $S\setminus \{ s_ a\} $, and choose $W\Subset V$ containing $s_ a$. There is a smooth function $\chi $ on $Y^\circ $, equal to one on $W\setminus S$ and zero outside $V$, whose derivative has compact support in $Y^\circ $. Indeed the transition takes place in the compact regular set $\overline V\setminus W$, where smooth cutoffs exist. Therefore $|\mathrm{d}\chi |_{g_0}\leq C$ for some finite $C$. Integration along curves gives
\[ |\chi (p)-\chi (q)|\leq C d_0(p,q),\qquad p,q\in Y^\circ . \]
The Lipschitz extension of $\chi $ to the completion has value one at $\widehat s_ a$ and zero at the regular point $y$, or at $\widehat s_ b$ if $y=s_ b$. This proves the assertion.
Define $\iota $ by the identity on the regular locus and by $\iota (s_ a)=\widehat s_ a$. The intrinsic distance induces the smooth topology at each regular point. At $s_ a$, adjoining $\widehat s_ a$ to a punctured neighborhood does not increase its diameter, so the diameter hypothesis proves continuity of $\iota $. Every completion point is represented by a Cauchy sequence in $Y^\circ $. Compact metrizability of $Y$ gives a subsequence converging topologically either to a regular point or to some $s_ a$; the preceding arguments give its metric limit. Hence $\iota $ is surjective as well as injective. A continuous bijection from compact $Y$ to Hausdorff $\widehat Y$ is a homeomorphism.
$\square$
Let $X$ be a compact connected smooth manifold and let $\pi :X\to Y$ be a continuous surjection, with $E=\pi ^{-1}(S)$ and with $\pi :X\setminus E\to Y^\circ $ a diffeomorphism. Assume that the identity of $Y^\circ $ extends to a homeomorphism $\iota :Y\to \widehat Y$, for instance under the hypothesis of lemma 15.19.21, and write $G=\iota \circ \pi $. For a smooth Riemannian metric $g$ on $X$ and an open subset $U$, let $\operatorname {diam}^{\mathrm{int}}(U,g)$ denote the intrinsic diameter of $U$, computed with the lengths of piecewise smooth curves contained in $U$.
Proposition 15.19.22.65049DB7855E Let $g_ j$ be smooth Riemannian metrics on $X$ such that $g_ j\to \pi ^*g_0$ uniformly on every compact subset of $X\setminus E$. Suppose each $s_ a$ has a nested neighborhood basis $V_{a,r}$, with $U_{a,r}=\pi ^{-1}(V_{a,r})$, satisfying
\[ \lim _{r\downarrow 0}\operatorname {diam}_{\widehat d_0}(\iota (V_{a,r}))=0, \qquad \lim _{r\downarrow 0}\limsup _{j\to \infty } \operatorname {diam}^{\mathrm{int}}(U_{a,r},g_ j)=0. \]
Then, for the global length distances $d_ j$ of $g_ j$,
\[ \lim _{j\to \infty }\sup _{x,x'\in X} |d_ j(x,x')-\widehat d_0(G(x),G(x'))|=0. \]
Consequently $(X,d_ j)$ converges to $(\widehat Y,\widehat d_0)$ in Gromov–Hausdorff distance.
Proof.
If $S$ is empty, uniform metric comparison on compact $X$ proves the distance convergence. Assume $N\geq 1$. Fix $\tau >0$. Choose pairwise disjoint neighborhoods $V_ a$ from the specified bases with
\[ \operatorname {diam}_{\widehat d_0}(\iota (V_ a))<\tau , \qquad \operatorname {diam}^{\mathrm{int}}(U_ a,g_ j)<\tau \]
for all sufficiently large $j$, where $U_ a=\pi ^{-1}(V_ a)$. Pick $q_ a\in V_ a\setminus S$, and then choose open neighborhoods $W_ a$ of $s_ a$ with $\overline W_ a\subset V_ a$ and $q_ a\notin \overline W_ a$. The regular cores
\[ K_ Y=Y\setminus \bigcup _ a W_ a, \qquad K_ X=\pi ^{-1}(K_ Y) \]
are compact and identified by $\pi $. Uniform convergence and positivity of $g_0$ on this core give numbers $A_ j\geq 1$, with $A_ j\to 1$, such that lengths of curves contained in the cores compare in both directions by the factor $A_ j$.
Consider a piecewise smooth path in $Y^\circ $ joining two points of $K_ Y$. The preimages of the closed sets $\overline W_ a$ are compact subsets of its parameter interval. At its first visit to any such set, delete the portion from that visit through its last visit to the same set. Resume with the remaining tail. That set is never visited again; thus there are at most $N$ deleted portions. Each retained segment lies in $K_ Y$, and each deleted portion has endpoints in a single $\overline W_ a\subset V_ a$. The lifts of those endpoints can be joined in $U_ a$ with $g_ j$-length less than $\tau $, up to an arbitrarily small error. Applying this procedure to almost minimizing paths, comparing retained lengths, and letting the errors tend to zero gives, for $x,x'\in K_ X$,
\[ d_ j(x,x')\leq A_ j\widehat d_0(G(x),G(x'))+N\tau . \]
Conversely start with an almost minimizing $g_ j$-path in $X$ and apply the same deletion procedure to the closed sets $\pi ^{-1}(\overline W_ a)$. Project its retained segments, which avoid $E$, to $K_ Y$. The two images of each deleted portion have completion distance less than $\tau $. The triangle inequality in $\widehat Y$ therefore gives
\[ \widehat d_0(G(x),G(x'))\leq A_ j d_ j(x,x')+N\tau , \qquad x,x'\in K_ X. \]
For any $x\notin K_ X$ with $\pi (x)\in W_ a$, replace $x$ by $p=\pi ^{-1}(q_ a)\in U_ a\cap K_ X$. Then $d_ j(x,p)<\tau $ and $\widehat d_0(G(x),G(p))<\tau $. Leave points of $K_ X$ fixed. Replacing both endpoints extends the two inequalities to all $x,x'\in X$, with the same error
\[ C_ j=(N+2+2A_ j)\tau . \]
If $D=\operatorname {diam}(\widehat Y,\widehat d_0)$, the first inequality gives $\operatorname {diam}(X,d_ j)\leq A_ jD+C_ j$. The positive and negative distance errors are consequently bounded by
\[ (A_ j-1)D+C_ j, \qquad (A_ j-1)(A_ jD+C_ j)+C_ j, \]
respectively. Their upper limits are at most $(N+4)\tau $. Letting $\tau \downarrow 0$ proves uniform convergence.
The graph of $G$ is a correspondence between $X$ and $\widehat Y$, and its distortion $b_ j$ for the distances $d_ j$ and $\widehat d_0$ is the supremum in the statement. By lemma 15.19.20,
\[ d_{\mathrm{GH}}((X,d_ j),(\widehat Y,\widehat d_0))\leq b_ j/2\longrightarrow 0. \]
$\square$
For the standard nodal small resolutions of 15.19.8 with closed $\theta $, the smooth convergence of the whole family off $E$ in theorem 15.19.17 gives the uniform convergence on compact subsets of $X\setminus E$ required in proposition 15.19.22. What remains are the two conditions on the diameters of neighborhoods. The next estimates control the approximating neighborhoods by metric bounds along and normal to the exceptional curves.
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