論文の公開元へMetric measure foliations, product spaces, and their convergence
概要
In recent years, the geometry and analysis on metric measure spaces have actively been studied. Metric measure spaces typically appear as limit spaces
of Riemannian manifolds in the convergence/collapsing theory of Riemannian manifolds. The study of convergence of metric measure spaces is one of
central topics in the geometry and analysis on metric measure spaces. Convergence notions of metric measure spaces are formulated variously such as
measured Gromov-Hausdorff convergence.
In this thesis, we study mainly the following two notions: the metric
measure foliation and the product of metric measure spaces. Our purpose is
to comprehend the behavior of these notions with respect to convergence of
metric measure spaces. Especially, we are interested in the behavior in the
case for sequences of metric measure spaces whose dimensions are unbounded.
This thesis is based on [11, 12].
Our settings are described more precisely as follows. We call a triple
(X, dX , mX ) a metric measure space if (X, dX ) is a complete separable metric
space and mX a Borel probability measure on X. We sometimes say that
X is a metric measure space, in which case the metric and the measure of
X are respectively indicated by dX and mX . As convergence notions of such
spaces, !-convergence and concentration were introduced by Gromov in [9].
!-convergence is a stronger convergence than concentration, that is, all !convergent sequences of metric measure spaces concentrate. !-convergence
is based on a simple idea and is naively formulated as the convergence of
the distance function dX and the reference measure mX . On the other hand,
concentration is based on the concentration of measure phenomenon studied
by L´evy and V. Milman, and it is formulated as the convergence of the set, say
Lip1 (X), of all 1-Lipschitz functions on a space X, instead of the distance
function dX , and the reference measure mX . From these different ideas,
concentration admits the convergence of many non-trivial sequences of metric
measure spaces whose dimensions are unbounded. This is one of the most
important characteristic of concentration. ...
