Gluing of metric measure spaces and the heat equation with homogeneous Dirichlet boundary values

Abstract

The first part of this thesis deals with gluing together several copies of an open subset of a metric measure space along the complement. This construction results in a metric measure space. We identify the Cheeger energy and the heat flow on the glued space in terms of the corresponding objects of the underlying space. Surprisingly, the heat flow on the glued space can be expressed by using the heat flow on the underlying space and the heat flow on the open subset with <em>homogeneous Dirichlet boundary conditions</em>. This yields a possibility to deal with the Dirichlet heat flow in terms of optimal transport theory. When the glued space satisfies a lower bound on the Ricci curvature, we can infer a gradient estimate and an equivalent Bochner inequality for the Dirichlet heat flow.<br> As the Dirichlet heat flow does not preserve mass, we have to deal with measures of unequal masses. This makes the usual Kantorovich-Wasserstein metric useless. Instead, using a new heuristic particle interpretation for the Dirichlet heat flow that also uses <em>antiparticles</em>, we can assume the sum of particles and antiparticles to be constant and use the Kantorovich-Wasserstein metric on such sums. However, this only yields a semi-metric (i.e. the triangle inequality might not be satisfied). There is a standard way to define an induced metric from this, and we will even go a step further and define the induced length metric from it.<br> Another related metric is obtained by studying the one-point completion of the open subset; the added point will serve as a cemetery which makes it possible to view a subprobability measure on the open set as a probability on the one-point completion and thus using the Kantorovich-Wasserstein metric on this space.<br> Deriving some representation formulas in terms of other transport costs, we can compare these metrics and also clarify the relationship to weak convergence of measures. The most precise results are obtained in the case <em>p</em>=1. Again under the assumption that the glued space has a lower bound on the Ricci curvature, we get contraction results in various of these new metrics.