# Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense

Let $$\{X_i^n\}$$ be a sequence of smooth compact Riemannian $$n$$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $$X_i$$ is geodesically locally convex near its boundary (the latter assumption is equivalent that $$X_i$$ is an Alexandrov space). Assume $$\{X_i\}$$ converges in the Gromov-Hausdorff sense to a compact smooth Riemannian manifold $$X$$ with boundary and $$\dim X=n$$, i.e. there is no collapse.

Is it true that volume of the bondary $$vol_{n-1}(\partial X_i)$$ converges to $$vol_{n-1}(\partial X)$$? The case $$n=2$$ is already interesting to me.

Remarks. (1) By the Perelman stability theorem $$X_i$$ is homeomorphic to $$X$$ for $$i\gg 1$$.

(2) By Burago-Gromov-Perelman one has convergence of volumes of manifolds $$vol_n(X_i)\to vol_n(X)$$.

According to Theorem 1.2. in my "Applications of quasigeodesics and gradient curves", $$\partial X_i\to \partial X$$ as length spaces in the sense of Gromov--Hausdorff. Then the same argument as in Burago--Gromov--Perelman shows that $$\mathrm{vol}\,\partial X_i\to \mathrm{vol}\,\partial X$$.
• Are the following statements true? Let $\{X_i^n\}$ be sequence of compact Alexandrov spaces with curvature uniformly bounded below which GH-converges to $X$ without collapse. Let $F_i\subset X_i$ be extremal subsets (e.g. boundaries) which converge is Hausdorff sense to $F\subset X$. Then: (1) $F$ is extremal subset of $X$. (2) $F_i\to F$ is GH-sense. (3) For large $i$ all $F_i$ and $F$ have the same Hausdorff dimension which is necessarily integer $k$. (4) $vol_k(F_i)\to vol_k(F)$ where $vol_k$ is the $k$th Hausdorff measure which is necessarily finite and positive for $F_i$ and $F$.
• @MKO (1) yes --- it follows since F is ideal of gradient flow, (2) For sure yes, moreover they converge as length-spaces, (3) yes, I am not sure if it stated this way somewhere, but there is a way to estimate volume of ambient space (better say $R$-nbhd of $F$ in the ambient space via area of $F$); hence if $F_i$ is collapsing, than so is $X_i$; (4) I already said yes (did I?). Commented Nov 19, 2019 at 18:38