Gromov's compactness theorem (geometry)
   HOME

TheInfoList



OR:

In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
metric geometry In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, Mikhael Gromov proved a fundamental compactness theorem for sequences of
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s. In the special case of
Riemannian manifolds In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
, the key assumption of his compactness theorem is automatically satisfied under an assumption on
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
. These theorems have been widely used in the fields of
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
and
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
.


Metric compactness theorem

The Gromov–Hausdorff distance defines a notion of distance between any two
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as Gromov–Hausdorff convergence. Gromov found a condition on a sequence of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
metric spaces which ensures that a
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
converges to some metric space relative to the Gromov–Hausdorff distance:
Let be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number there is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
and, for every , the set can be covered by metric balls of radius . Then the sequence has a subsequence which converges relative to the Gromov–Hausdorff distance.
The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the
Arzelà–Ascoli theorem The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interv ...
in the theory of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
. Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
, where he applied it to define the
asymptotic cone In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and use ...
of certain metric spaces. These techniques were later extended by Gromov and others, using the theory of
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
s.


Riemannian compactness theorem

Specializing to the setting of geodesically complete Riemannian manifolds with a fixed lower bound on the
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
, the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem. As such, it follows that:
Consider a sequence of closed Riemannian manifolds with a uniform lower bound on the Ricci curvature and a uniform upper bound on the diameter. Then there is a subsequence which converges relative to the Gromov–Hausdorff distance.
The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure. This special case of the metric compactness theorem is significant in the field of
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, as it isolates the purely metric consequences of lower Ricci curvature bounds.


References

Sources. * * * * * * * {{Manifolds Differential geometry Theorems in Riemannian geometry