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The heat flow on glued spaces with varying dimension

Anton Ullrich

TL;DR

The work develops a diffusion framework on glued spaces formed by manifolds of varying dimension connected by weighted measures. It constructs a canonical heat flow via a Dirichlet form $\mathcal{E}(f)=\int_X |\nabla f|^2 \, d\mu$ and proves ergodicity and irreducibility under an $N$-doubling condition and transversal intersections, yielding a spectral gap. Capacity bounds and a Rellich–Kondrachov embedding are key technical tools that ensure the intersection remains connected and drive convergence of the flow to the mean, with Brownian motion interpretation arising from the Laplace–Beltrami generator $\mathcal{A}$. The results extend to multiple manifolds and general weights, and the paper outlines a program for a $\Gamma$-convergence of a nonlocal heat-excess perimeter to the local perimeter, connecting diffusion geometry on glued spaces to BV and learning applications.

Abstract

In this paper, we introduce a new concept of glued manifolds and investigate under which conditions the canonical heat flow on these glued manifolds is ergodic and irreducible. Glued manifolds are metric spaces consisting of manifolds of varying dimension connected by a weakly doubling measure. This can be seen as a condition on the jump in dimension. From another perspective, this construction also defines the Brownian motion on these glued spaces. Using the heat flow, we construct a nonlocal perimeter functional, the heat excess, to raise the question of its $Γ$-convergence to the standard perimeter functional. In this context, we connect our work to the previous work on the convergence of perimeter functionals, approximations, and existence of heat kernels, as well as short-time expansions of Brownian motion.

The heat flow on glued spaces with varying dimension

TL;DR

The work develops a diffusion framework on glued spaces formed by manifolds of varying dimension connected by weighted measures. It constructs a canonical heat flow via a Dirichlet form and proves ergodicity and irreducibility under an -doubling condition and transversal intersections, yielding a spectral gap. Capacity bounds and a Rellich–Kondrachov embedding are key technical tools that ensure the intersection remains connected and drive convergence of the flow to the mean, with Brownian motion interpretation arising from the Laplace–Beltrami generator . The results extend to multiple manifolds and general weights, and the paper outlines a program for a -convergence of a nonlocal heat-excess perimeter to the local perimeter, connecting diffusion geometry on glued spaces to BV and learning applications.

Abstract

In this paper, we introduce a new concept of glued manifolds and investigate under which conditions the canonical heat flow on these glued manifolds is ergodic and irreducible. Glued manifolds are metric spaces consisting of manifolds of varying dimension connected by a weakly doubling measure. This can be seen as a condition on the jump in dimension. From another perspective, this construction also defines the Brownian motion on these glued spaces. Using the heat flow, we construct a nonlocal perimeter functional, the heat excess, to raise the question of its -convergence to the standard perimeter functional. In this context, we connect our work to the previous work on the convergence of perimeter functionals, approximations, and existence of heat kernels, as well as short-time expansions of Brownian motion.
Paper Structure (12 sections, 8 theorems, 87 equations, 3 figures)

This paper contains 12 sections, 8 theorems, 87 equations, 3 figures.

Table of Contents

  1. Motivation
  2. Structure of the paper
  3. Setting and Main Theorem
  4. Basic Theorems for A2-Weights on Manifolds
  5. Proof of Main Theorem
  6. Definition of the Heat Flow
  7. Capacity Bounds
  8. Compact Embedding of H1
  9. Generalizations
  10. Outlook
  11. Related work
  12. Acknowledgement

Key Result

Theorem 1

Let $(X,d,\mu)$ be the union of weighted smooth manifolds with transversal intersections such that $\mu$ satisfies a weak doubling condition. Then, the canonical heat flow in $X$ is ergodic and irreducible.

Figures (3)

  • Figure 1: Example of a glued manifold consisting of the weighted manifolds $M_1$ and $M_2$.
  • Figure 2: A sample path of the Brownian motion on a sphere.
  • Figure 3: We prove the lower capacity bound via a foliation procedure as shown in this figure.

Theorems & Definitions (41)

  • Theorem
  • Definition 2.1: Glued distance
  • Definition 2.2: Transversality, guillemin2010differential
  • Proposition 2.3: guillemin2010differential
  • Definition 2.4: $N$-doubling condition
  • Remark
  • Remark
  • Theorem 2.5: Heat flow on the glued manifold
  • Example 2.6
  • Definition 2.7: Dirichlet form
  • ...and 31 more