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Liouville theorems and removable sets for bounded $p$ -harmonic and quasiharmonic functions on metric spaces under local assumptions

This paper characterizes the removable compact sets for bounded $p$ -harmonic and quasiharmonic functions on metric spaces with local doubling and Poincaré conditions by establishing their equivalence to Liouville-type theorems and identifying key geometric and analytic properties such as local connectedness and $p$ -parabolicity.

Original authors: Anders Björn, Jana Björn

Published 2026-08-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Anders Björn, Jana Björn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a landscape where the rules of geometry are slightly different from the flat, familiar world we walk on every day. In this landscape, which mathematicians call a metric space, distance is measured not just by straight lines but by the paths one can actually travel. Within this world, there are special functions—mathematical descriptions of how quantities like heat or electric potential settle down—that behave in a very smooth, predictable way. These are the harmonic functions. They are the natural state of equilibrium, the way a stretched rubber sheet settles when you stop pulling it. For over a century, mathematicians have been fascinated by a specific question about these functions: if you punch a hole in the landscape, does the smoothness of the function break, or can it simply flow over the gap as if the hole were never there? This question of "removability" is crucial because it tells us whether a tiny flaw in the universe can change the fundamental laws governing it.

The answer depends heavily on the shape of the hole and the nature of the space itself. In the flat world of standard geometry, we know that very small holes, like a single point, often do not matter; the function can be extended across them without issue. However, if the hole is too large or the space is shaped in a way that traps energy, the smoothness breaks, and the function cannot be fixed. A team of researchers at Linköping University in Sweden has now mapped out exactly when this happens in a vast array of complex, non-flat landscapes. They studied spaces that are connected and have a specific type of measure, which is a way of assigning size or volume to regions, and which support a local version of a fundamental inequality that governs how functions change. Their goal was to determine precisely which compact sets—think of them as finite, closed islands of missing space—can be ignored by bounded functions that describe these equilibrium states.

The researchers discovered that the ability of a function to ignore a hole is not just about the size of the hole, but about the global character of the space and the local geometry around the hole's edge. They found that for a hole to be truly removable, the space must either be finite in extent or possess a specific "parabolic" quality, meaning it is large enough that energy cannot escape to infinity in a certain way. Furthermore, the hole itself must be very specific: it must be so small that it essentially consists of a single point of significance, with all other parts of its boundary being negligible. Most importantly, the space immediately surrounding the hole must be locally connected, meaning that if you stand near the hole, you can always find a path to any other nearby point without jumping over a gap. If the space is fragmented or disconnected right next to the hole, the function cannot cross over, and the hole remains a permanent scar.

One of the most striking findings is that these conditions are not just helpful hints; they are absolute requirements. The authors proved that if a space is not locally connected at the edge of a hole, no matter how small the hole is, there will always be a smooth function that gets stuck and cannot be extended. They also showed that if the space is infinite and has a "hyperbolic" nature, allowing energy to escape freely, then even a single point can act as an unfixable barrier unless the space has a very specific, rare structure. The team provided a complete checklist for determining removability: the space must be of a certain type, the hole must be concentrated at a single point, and the function must approach a clear, single value as it gets closer to that point. If any of these conditions fail, the hole is permanent.

To reach these conclusions, the researchers developed a new, elementary proof for a famous theorem known as the Liouville theorem, which states that in certain types of spaces, the only smooth functions that stay within a fixed range are constant ones. They showed that this rule holds even under very local assumptions, without needing the entire space to be uniform. This allowed them to treat the problem of removable holes as a test of whether the space forces all bounded functions to be constant. If the space forces the function to be constant, then the hole is removable because a constant function can easily be extended across any gap. If the space allows for non-constant functions, then the hole might be a barrier.

The paper also explores the limits of these rules through a series of carefully constructed examples. They built mathematical models that look like twisted shapes or networks of lines to show that intuition can be misleading. For instance, they demonstrated that a space can be locally connected—meaning you can walk around the hole without getting stuck—yet still fail to allow a function to cross it because the paths are too long or winding in a specific way. Conversely, they showed that in some infinite spaces, even a set that separates the space into two pieces can be removable, provided the space has the right parabolic properties. These examples serve as a warning against assuming that a single geometric feature, like connectedness, is enough to guarantee removability.

Ultimately, this work provides a definitive guide for mathematicians working in fields ranging from the study of curved surfaces to the analysis of weighted spaces where the "volume" of a region changes from point to point. It clarifies that the removability of a singularity is a delicate balance between the global nature of the universe and the local geometry of the defect. The results confirm that while some holes are merely illusions that smooth functions can glide over, others are fundamental breaks in the fabric of the space that no amount of mathematical smoothing can repair. The researchers have drawn the line in the sand, showing exactly where the smooth world ends and the broken world begins.

Technical Summary: Liouville theorems and removable sets for bounded p-harmonic and quasiharmonic functions on metric spaces under local assumptions

Problem Statement
This paper investigates the removability of singularities for bounded $p$ -harmonic and quasiharmonic functions within the context of general metric spaces. Specifically, the authors consider a proper, connected metric space $X$ equipped with a locally doubling measure $\mu$ that supports a local $p$ -Poincaré inequality ( $1 < p < \infty$ ). The central question is to characterize which compact sets $K \subset \Omega$ (where $\Omega \subset X$ is open) with positive $p$ -capacity ( $C_p(K) > 0$ ) are "removable." A set $K$ is defined as removable if every bounded $p$ -harmonic (or $Q$ -quasiharmonic) function defined on $\Omega \setminus K$ admits a bounded extension to $\Omega$ that remains $p$ -harmonic (or $Q$ -quasiharmonic).

While it is well-established that sets with zero capacity are always removable, the behavior of sets with positive capacity is more complex. The paper addresses the gap in characterizing these sets under local geometric and analytic assumptions, rather than the global assumptions often found in classical literature.

Methodology
The authors employ tools from first-order analysis on metric spaces, specifically utilizing Newtonian Sobolev spaces ( $N^{1,p}$ ), upper gradients, and the theory of $p$ -harmonic functions developed in the context of metric measure spaces. The methodology relies on:

Dirichlet Problem Solutions: The authors utilize both Sobolev solutions (minimizers of the $p$ -energy integral) and Perron solutions (constructed via superharmonic barriers) to analyze boundary behavior and extensions.
Capacity Theory: The analysis heavily depends on the $p$ -capacity ( $C_p$ ) and condenser capacity to distinguish between removable and non-removable sets.
Topological and Geometric Conditions: The paper introduces and leverages specific topological properties, including:
- Local connectedness: The connectivity of the domain near boundary points.
- Sequential annular quasiconvexity: A condition ensuring that points on spheres around a singularity can be connected by curves within annular regions of the domain.
- Concentration of capacity: The property that the capacity of the boundary of the domain is concentrated at a single point.
Liouville Theorems: A significant portion of the methodology involves proving Liouville-type theorems (stating that bounded entire functions must be constant) for quasisuperharmonic functions in $p$ -parabolic spaces. The authors provide an elementary proof for this theorem that relies solely on the definition of $p$ -parabolicity, avoiding the weak Harnack inequality typically required under global assumptions.

Key Contributions and Results

Characterization of Removable Sets (Theorem 1.3): The primary result provides a complete characterization of compact sets $K$ with $C_p(K) > 0$ that are removable for bounded $Q$ -quasiharmonic functions. The removability of $K$ in $G = \Omega \setminus K$ is shown to be equivalent to the following conditions:
1. $X$ is either $p$ -parabolic or bounded.
2. There exists a point $x_0 \in \partial K$ such that $C_p(\partial G \setminus \{x_0\}) = 0$ .
3. The limit $\lim_{G \ni y \to x_0} u(y)$ exists for every bounded $Q$ -quasiharmonic function $u$ in $G$ .
  Crucially, the paper establishes that these conditions are equivalent to the validity of a Liouville theorem in $G$ (i.e., all bounded $Q$ -quasiharmonic functions in $G$ are constant).
Liouville Theorem in $p$ -Parabolic Spaces (Theorem 1.8): The authors prove that in a proper connected metric space that is $p$ -parabolic or bounded, every nonnegative quasisuperharmonic function is constant. This result is derived using only the definition of $p$ -parabolicity and properties of quasisuperharmonic functions, offering a more elementary approach than previous proofs relying on the weak Harnack inequality.
Necessity of Local Connectedness (Proposition 1.7): The paper demonstrates that local connectedness at the boundary of the singularity is a necessary condition for removability. If a set $E$ is not locally connected at a boundary point $x_0$ , there exists a bounded $p$ -harmonic function in the complement that cannot be continuously extended to $x_0$ .
Sequential Annular Quasiconvexity (Lemma 9.2): The authors show that if a domain is sequentially annularly quasiconvex at a boundary point $x_0$ , then the limit of any bounded quasiharmonic function at $x_0$ exists. This geometric condition is identified as a sufficient condition for the existence of limits, bridging the gap between geometric structure and analytic behavior.
Counterexamples and Limitations: The paper provides several examples (Sections 10 and 11) illustrating that:
- Removability does not hold if the compact set assumption is replaced by unbounded closed sets (Theorem 1.3 fails in this case).
- Local connectedness is not sufficient for the existence of limits; sequential annular quasiconvexity is strictly stronger (Examples 11.2 and 11.3).
- In $p$ -hyperbolic spaces, sets with positive capacity can be removable only under very specific geometric configurations (e.g., "bow-tie" geometries).

Significance and Scope
The paper claims significance in extending the theory of removable singularities from weighted Euclidean spaces ( $\mathbb{R}^n$ ) to general metric spaces under local assumptions. By characterizing removability in terms of geometric properties (like sequential annular quasiconvexity) and analytic properties (Liouville theorems), the authors unify the understanding of these phenomena across manifolds and spaces with weights.

The results apply specifically to:

Connected Riemannian and Finsler manifolds (with or without boundary) equipped with locally doubling measures supporting local $p$ -Poincaré inequalities.
Spaces equipped with $p$ -admissible weights.

The authors modestly note that while they provide a complete characterization for compact sets, the behavior of unbounded closed sets remains more complex and requires different conditions, as demonstrated by their counterexamples. Furthermore, they identify sequential annular quasiconvexity as a crucial, previously underutilized condition for studying removable sets in metric spaces. The work refines the understanding of the relationship between the geometry of the space, the capacity of the singularity, and the analytic properties of $p$ -harmonic functions.

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