Gromov-Hausdorff convergence

The Hausdorff distance {d_{ H}(A,B)} between two subsets {A,B} of a metric space {X} is defined by {d_{ H}(A,B)=\inf\{r>0: A\subset B_r \text{ and } B\subset A_r \}}, where {A_r,B_r} are (open/closed does not matter) {r}-neighborhoods of the sets. Informally: {d_{ H}(A,B)<r} if no matter where you are in one set, you can jump into the other by traveling less than {r}. For example, the distance between letters S and U is about the length of the longer green arrow.

Hausdorff distance
Hausdorff distance

Generally, one assumes the sets to be closed (to avoid zero distance) and bounded (to avoid infinite distance). But I will not; in this post I’m not really interested in verifying all the axioms of a metric.

The Gromov-Hausdorff distance is defined between metric spaces {X,Y} as follows: it is the infimum of {d_{ H}(f(X),g(Y))} taken over all isometric embeddings {f\colon X\rightarrow Z} and {g\colon Y\rightarrow Z} into some metric space {Z}.

The infimum over all pairs of embeddings into all conceivable metric spaces does not sound like something you would want to compute in practice. Of course, the matter boils down to equipping the abstract union {X\sqcup Y} with pseudometrics that are compatible with the original metrics on {X} and {Y}.

A more directly computable notion of distance (not necessarily a metric) can be given as follows: {\rho_{GH}(X,Y)} is the infimum of all {\epsilon>0} for which there exist two maps {f\colon X\rightarrow Y} and {g\colon Y\rightarrow X} such that:

  1. {d_X(g\circ f(x),x)\le \epsilon} for all {x\in X}
  2. {d_Y(f\circ g(y),y) \le \epsilon} for all {y\in Y}
  3. {|d_Y(f(x_1), f(x_2)) - d_X(x_1,x_2)| \le \epsilon } for all {x_1,x_2\in X}
  4. {|d_X(g(y_1), g(y_2)) - d_Y(y_1,y_2)| \le \epsilon } for all {y_1,y_2\in Y}

This is not as elegant as “infimize over all metric space”, but is more practical. For example, it is easy to check that the sequence of one-sheeted hyperboloids {H_n = \{x^2+y^2=z^2+1/n\}}

One hyperboloid,...
One hyperboloid,…
... another hyperboloid, ...
… another hyperboloid, …

converges to the cone {C = \{x^2+y^2=z^2\}}.

... and we get a cone.
… and we get a cone.

Using cylindrical coordinates, define {f_n\colon H_n\rightarrow C} by {f_n(r,\theta,z) = (\sqrt{r^2-1/n}, \theta,z) } and {g\colon C\rightarrow H_n} by {g_n(r,\theta,z) = (\sqrt{r^2+1/n}, \theta,z)}, with an arbitrary choice of {\theta} at the point {g(0,0,0)}. Now check the items one by one:

  1. {f_n\circ g_n} is the identity map on {C}
  2. {g_n\circ f_n} fixes all points of {H_n} except for those with {z=0}. The latter are displaced by at most {2/\sqrt{n}}.
  3. follows from {|\sqrt{r^2-1/n} - r|\le 1/\sqrt{n}} on {[1/n,\infty)}
  4. follows from {|\sqrt{r^2+1/n} - r|\le 1/\sqrt{n}} on {[0,\infty)}

Note that the Gromov-Hausdorff convergence of manifolds is understood with respect to their intrinsic metrics. Although both {H_n} and {C} are naturally identified with subsets of {\mathbb R^3}, it would be a mistake to use the Hausdorff distance based on the Euclidean metric of {\mathbb R^3}. Even though this extrinsic metric is bi-Lipschitz equivalent to the intrinsic metric on both {H_n} and {C}, bi-Lipschitz equivalence is too coarse to preserve the GH convergence in general. In general, the intrinsic metric on a manifold cannot be realized as the extrinsic metric in any Euclidean space.

In the example of hyperboloids we are lucky to have a Gromov-Hausdorff convergent sequence of unbounded spaces. Normally, bounded sets are required, and boundedness is imposed either by an exhaustion argument, or by changing the metric (moving to the projective space). For instance, the parabolas {y=x^2/n} do not converge to the line {y=0} as unbounded subsets of {\mathbb R^2}, but they do converge as subsets of {\mathbb R\mathbb P^2}. In the process, the degree jumps down from {2} to {1}.

It seems that the Gromov-Hausdorff limit of algebraic varieties of degree at most {d} is also an algebraic variety of degree at most {d} (provided that we use the intrinsic metric; otherwise flat ellipses {x^2+n^2y^2=1} would converge to a line segment). If this is true, I’m sure there’s a proof somewhere but I never saw one.

1 thought on “Gromov-Hausdorff convergence”

  1. What about the family of higher-order cusps y^2=x^n? Algebraically, the n in the exponent makes that term get very small (“in the adic topology”). Is it clear that these converge to y^2=0? (Of course none of these are manifolds, but that didn’t seem to bother YY at all.)

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