Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semicontinuity of capacity under pointed intrinsic flat convergence

Jeffrey L. Jauregui, Raquel Perales, Jacobus W. Portegies

TL;DR

The paper investigates how the capacity of a compact set, defined via Dirichlet energy of Lipschitz competitors, behaves when the ambient space varies through pointed volume-preserving intrinsic flat convergence. By developing an extrinsic, ambient-space framework and introducing corresponding regions, the authors prove two upper semicontinuity results: (i) for balls of fixed radius around converging base points, and (ii) for Lipschitz sublevel sets, under pointed VF convergence. The results are supported by a detailed construction of corresponding regions and energy-control arguments, and are shown to be sharp through several examples that highlight the necessity of the VF hypothesis and illustrate possible jumps. The work connects capacity semicontinuity to non-smooth notions of mass in general relativity, offering a potential pathway toward defining and understanding total mass in AF spaces that lack smooth structures, via the capacity-volume mass $m_{CV}$ and related notions.

Abstract

The concept of the capacity of a compact set in $\mathbb R^n$ generalizes readily to noncompact Riemannian manifolds and, with more substantial work, to metric spaces (where multiple natural definitions of capacity are possible). Motivated by analytic and geometric considerations, and in particular Jauregui's definition of capacity-volume mass and Jauregui and Lee's results on the lower semicontinuity of the ADM mass and Huisken's isoperimetric mass, we investigate how the capacity functional behaves when the background spaces vary. Specifically, we allow the background spaces to consist of a sequence of local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. For the case of volume-preserving ($\mathcal{VF}$) convergence, we prove two theorems that demonstrate an upper semicontinuity phenomenon for the capacity: one version is for balls of a fixed radius centered about converging points; the other is for Lipschitz sublevel sets. Our approach is motivated by Portegies' investigation of the semicontinuity of eigenvalues under $\mathcal{VF}$ convergence. We include examples to show the semicontinuity may be strict, and that the volume-preserving hypothesis is necessary. Finally, there is a discussion on how capacity and our results may be used towards understanding the general relativistic total mass in non-smooth settings.

Semicontinuity of capacity under pointed intrinsic flat convergence

TL;DR

The paper investigates how the capacity of a compact set, defined via Dirichlet energy of Lipschitz competitors, behaves when the ambient space varies through pointed volume-preserving intrinsic flat convergence. By developing an extrinsic, ambient-space framework and introducing corresponding regions, the authors prove two upper semicontinuity results: (i) for balls of fixed radius around converging base points, and (ii) for Lipschitz sublevel sets, under pointed VF convergence. The results are supported by a detailed construction of corresponding regions and energy-control arguments, and are shown to be sharp through several examples that highlight the necessity of the VF hypothesis and illustrate possible jumps. The work connects capacity semicontinuity to non-smooth notions of mass in general relativity, offering a potential pathway toward defining and understanding total mass in AF spaces that lack smooth structures, via the capacity-volume mass and related notions.

Abstract

The concept of the capacity of a compact set in generalizes readily to noncompact Riemannian manifolds and, with more substantial work, to metric spaces (where multiple natural definitions of capacity are possible). Motivated by analytic and geometric considerations, and in particular Jauregui's definition of capacity-volume mass and Jauregui and Lee's results on the lower semicontinuity of the ADM mass and Huisken's isoperimetric mass, we investigate how the capacity functional behaves when the background spaces vary. Specifically, we allow the background spaces to consist of a sequence of local integral current spaces converging in the pointed Sormani--Wenger intrinsic flat sense. For the case of volume-preserving () convergence, we prove two theorems that demonstrate an upper semicontinuity phenomenon for the capacity: one version is for balls of a fixed radius centered about converging points; the other is for Lipschitz sublevel sets. Our approach is motivated by Portegies' investigation of the semicontinuity of eigenvalues under convergence. We include examples to show the semicontinuity may be strict, and that the volume-preserving hypothesis is necessary. Finally, there is a discussion on how capacity and our results may be used towards understanding the general relativistic total mass in non-smooth settings.
Paper Structure (15 sections, 20 theorems, 177 equations, 4 figures)

This paper contains 15 sections, 20 theorems, 177 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definitions and basic objects
  3. Locally integral currents
  4. Flat and weak convergence of integral currents
  5. Local integral current spaces and pointed $\mathcal{F}$-convergence
  6. Dirichlet energy and capacity
  7. Semicontinuity of capacity
  8. Corresponding regions
  9. Main results and proofs
  10. Examples
  11. Example 1: Transition from cylindrical to Euclidean end geometry
  12. Example 2: Formation of a new end
  13. Example 3: Capacity jump with $\mathcal{VF}$-convergence
  14. Example 4: Cancellation and necessity of volume-preserving $\mathcal{F}$ convergence
  15. Asymptotically flat local integral current spaces and general relativistic mass

Key Result

Theorem 1

Let $N_i = (X_i, d_i, T_i)$ and $N_\infty = (X_\infty, d_\infty, T_\infty)$ be local integral current spaces of dimension $m \geq 2$, such that $N_i \to N_\infty$ in the pointed volume-preserving intrinsic flat sense with respect to $p_i \in X_i$ and $p_\infty \in X_\infty$. Suppose the closed ball

Figures (4)

  • Figure 1: The sets $K \subset S =\mathop{\mathrm{set}}\nolimits(T) \subset Y$ are defined in Theorem \ref{['thm_extrinsic']}. The function $f: S \to \mathbb{R}$ from Proposition \ref{['prop-Energy']} is identically $1$ on $O \subset S$ which contains $K$. In Lemma \ref{['lem-extensionf']}, $\gamma>0$ is chosen so that $K \subseteq K^{3\gamma} \subseteq O$, and $f$ is extended as a constant $1$ to $U^{-1}(-\infty, \gamma] \subset Y$. In Lemma \ref{['lem:Kdeltas']}, we introduce open balls $W \subset W'\subset Y$.
  • Figure 2: The finitely many sets $A_\ell$ are subsets of $S'\setminus K^\delta$ and are defined in Lemma \ref{['lem:setsAandV']}. They cover all of $S'\setminus K^\delta$ except for $\|T\|$-measure $< \epsilon_4$. The $V_\ell \subset Y$ are disjoint neighborhoods of the $A_\ell$, each of which lies in $W$ or $Y \setminus \overline W$. The larger ball $W'$ is not pictured, but $S' = S \cap W'$.
  • Figure 3: In Example 3, the space $X_i$ is the union of a unit 3-ball $K$ and a 3-space minus a unit ball, sitting at height $\frac{1}{i}$ above $K$ in $\mathbb{R}^4$. $X_i$, naturally viewed as a local integral current space, pointed $\mathcal{VF}$-converges to $X$ with respect to the origin, which is Euclidean 3-space. The corresponding regions for $K$ are simply $K_i=K$.
  • Figure 4: In Example 4, the space $X_i$ is the union of a hyperplane with an oppositely-oriented annular region sitting above at height $\frac{1}{i}$ in $\mathbb{R}^4$. $X_i$, naturally viewed as a local integral current space, converges in the pointed $\mathcal{F}$-sense (but not $\mathcal{VF}$) to $X$, which is Euclidean 3-space minus an annular region representing where the cancellation occurred.

Theorems & Definitions (53)

  • Theorem 1
  • Definition 2: LW
  • Definition 3: LW
  • Definition 4: LW
  • Definition 5: LW
  • Lemma 6
  • Definition 7: SW
  • Definition 8: SW
  • Definition 9: SW, Sor2
  • Theorem 10: Theorem 4.2 of SW
  • ...and 43 more