Length spaces with continuous length functional: is this set Gromov-Hausdorff closed? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:58:24Z http://mathoverflow.net/feeds/question/83961 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83961/length-spaces-with-continuous-length-functional-is-this-set-gromov-hausdorff-cl Length spaces with continuous length functional: is this set Gromov-Hausdorff closed? macbeth 2011-12-20T17:02:34Z 2011-12-23T15:23:28Z <p>As far as I can tell, a major motivation for the study of <a href="http://en.wikipedia.org/wiki/Intrinsic_metric" rel="nofollow">length spaces</a> is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,</p> <ol> <li>A complete connected Riemannian manifold is a complete length space.</li> <li>A Gromov-Hausdorff limit of complete length spaces is a complete length space.</li> </ol> <p>But of course there are stronger metric properties of Riemannian manifolds that one might hope would carry over to their limits. One that I have been wondering about is the continuity (say in the <s>compact-open topology</s> <strong>EDIT (see below):</strong> some other topology) of the length functional. After a couple of days' thought I've decided I have absolutely no intuition for this. So, I'd be very glad to hear:</p> <ol> <li>Is the length functional of a complete connected Riemannian manifold indeed continuous? (Proof in some special cases: if $\Gamma:[0,1]\times (-\epsilon,\epsilon)\to M$ is continuously differentiable, then $$\lim_{t\to 0}\int_0^1|\frac{\partial\Gamma}{\partial s}(s,t)|ds = \int_0^1|\frac{\partial\Gamma}{\partial s}(s,0)|ds$$ by limit-swapping.)</li> <li>Is a Gromov-Hausdorff limit of complete-length-spaces-with-continuous-length-functional also a complete-length-space-with-continuous-length-functional?</li> </ol> <hr> <p><strong>EDIT:</strong> It was quickly pointed out by Anton Petrunin, Pietro Majer and Vitali Kapovitch that for the compact-open topology, the answer to the first question is no (and that the second question is vacuous). Is it possible that there is some finer topology on (perhaps some subspace of) the space of curves in a length space, for which the answer to these questions is yes?</p> <p>For instance, consider the following property that a length space $X$ (with length functional $\mathcal{L}$) might possess:</p> <blockquote> <p>For any Lipschitz map $\Gamma:[0,1]\times(-\epsilon,\epsilon)\to X$, $$\lim_{t\to 0}\ \mathcal{L}(\Gamma(\cdot,t))=\mathcal{L}(\Gamma(\cdot,0)).$$</p> </blockquote> <p>It seems plausible to me that this would be true of complete connected Riemannian manifolds and that it would not be true of arbitrary length spaces. Is this so? And if so, is the set of length spaces which do have this property Gromov-Hausdorff closed?</p> http://mathoverflow.net/questions/83961/length-spaces-with-continuous-length-functional-is-this-set-gromov-hausdorff-cl/84149#84149 Answer by Marius Buliga for Length spaces with continuous length functional: is this set Gromov-Hausdorff closed? Marius Buliga 2011-12-23T09:02:07Z 2011-12-23T15:23:28Z <p>In a metric space $(X,d)$, you may look at the length functional as a FUNCTIONAL defined over the space of Lipschitz functions from $[0,1]$ to $(X,d)$. One of the natural notions of convergence of functionals is the <a href="http://en.wikipedia.org/wiki/%25CE%2593-convergence" rel="nofollow">gamma-convergence</a>. </p> <p>For a sequence of metric spaces $(X_{n},d_{n})$ which converges (in the Gromov-Hausdorff distance) to a metric space $(X,d)$, we know that there is a metric space $(Y,D)$, a sequence of isometric embeddings $F_{n}: (X,d_{n}) \rightarrow (Y,D)$ and another isometric embedding $F:(X,d) \rightarrow (Y,D)$, such that the sequence $F_{n}(X_{n})$ Hausdorff converges to $F(X)$. </p> <p>A natural question would be: is there a sequence of isometries $f_{n}: (X,d_{n}) \rightarrow (Y,D_{n})$ and an isometry $f:(X,d) \rightarrow (Y,D)$, such that</p> <ul> <li><p>$D_{n}$ and $D$ generate the same <a href="http://en.wikipedia.org/wiki/Uniformity_%2528topology%2529" rel="nofollow">uniformity</a> on $Y$,</p></li> <li><p>$D_{n}$ converges uniformly (on compact sets, on bounded sets, etc, pick your choice) to $D$,</p></li> <li><p>pick the topology of uniform convergence on the space $\mathcal{C}([0,1],Y)$. Then the sequence of length functionals associated to $(f_{n}(X_{n}), D_{n})$ gamma converges to the length functional of the limit $(f(X), D)$. </p></li> </ul> <p>If true then any sequence of $D_{n}$ geodesics (length minimizers) converges to a geodesic of $D$, for example. </p> <p>This is true in a Riemannian manifold (seen as the metric space $(X,d)$), if you take $(X_{n},d_{n})$ as $(B(x,1/n), \frac{1}{n}d)$, which GH converges to the (unit ball in the) tangent space at $x \in X$, for every point $x$. (This follows from results from the paper G. Buttazzo, L. De Pascale, I. Fragala, Topological equivalence of some variational problems involving distances, Discrete Contin. Dynam. Systems 7 (2001), no. 2, 247-258). </p> <p>It is also true for sub-riemannian manifolds, see the paper M. Buliga, A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111, <a href="http://arxiv.org/abs/0810.5042" rel="nofollow">arxiv link</a>. </p> <p>All this is related to the metric characterization of riemannian (and sub-riemannian) spaces, but in a different way than the great paper I.G. Nikolaev, A metric characterization of riemannian spaces, Siberian Advances in Mathematics, 1999, v. 9, N4, 1-58, <a href="http://www.ams.org/mathscinet-getitem?mr=1749850" rel="nofollow">MathSciNet link</a>. </p>