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arxiv: 2606.24397 · v1 · pith:QIKXCMGRnew · submitted 2026-06-23 · 🧮 math.AP · math.OC

Obstacles and Singularities of Riemannian Distance Functions

Paolo Albano , Piermarco Cannarsa , Carlo Sinestrari This is my paper

Pith reviewed 2026-06-25 23:49 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords Riemannian distancesingularitiesobstaclesHamilton-Jacobi equationlevel setspropagationcut locus
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The pith

The obstacle in a Riemannian manifold generates singularities in the distance function from a point, appearing in every high level set and propagating along Lipschitz arcs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the distance function from a fixed point in a Riemannian manifold that contains a compact obstacle. It shows that the obstacle forces the creation of singularities in this distance function. Specifically, every level set beyond a certain value includes at least one singular point. Singular points away from the obstacle are connected via nontrivial Lipschitz arcs. This extends known propagation results for singularities in Hamilton-Jacobi equations to the case with obstacles, and examples confirm the results cannot be improved much further.

Core claim

We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton--Jacobi equations.

What carries the argument

The Riemannian distance function from a point target avoiding a compact obstacle, whose level sets and singularities are studied using the associated Hamilton-Jacobi equation.

If this is right

  • Every sufficiently high level set of the distance function contains a singular point.
  • Singular points outside the obstacle form part of a Lipschitz arc of singularities.
  • The results hold for any smooth Riemannian metric and any compact obstacle not containing the target.
  • Examples exist where the distance function is differentiable at all boundary points of a nonconvex obstacle.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unavoidable singularities could affect numerical methods for computing distances or geodesics in domains with obstacles.
  • The Lipschitz arc property may help in classifying the structure of the singular set in more general settings.
  • One could test if similar results hold when the metric is only continuous or the obstacle has more complex topology.

Load-bearing premise

The Riemannian metric is smooth and the obstacle is compact and does not include the target point.

What would settle it

A counterexample consisting of a smooth Riemannian manifold and compact obstacle where there exists an arbitrarily large level set of the distance function with no singular points would falsify the claim.

read the original abstract

We study the distance function from a point target in the complement of a compact obstacle endowed with a smooth Riemannian metric. We prove that the obstacle necessarily generates singularities of the distance function: every sufficiently high level set contains a singular point. We also show that every singular point outside the obstacle belongs to a nontrivial Lipschitz arc of singularities, thereby extending to the constrained setting classical propagation results for Hamilton--Jacobi equations. Finally, we provide examples showing that these results are essentially sharp, including a nonconvex obstacle for which the distance function is differentiable at every boundary point.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the distance function from a fixed point target to points in the complement of a compact obstacle on a smooth Riemannian manifold. It proves that the obstacle necessarily generates singularities, in that every sufficiently high level set of the distance function contains at least one singular point, and that every singular point outside the obstacle lies on a nontrivial Lipschitz arc of singularities. The results are shown to be sharp by explicit examples, including a nonconvex obstacle on which the distance function remains differentiable at all boundary points.

Significance. If the central arguments hold, the work extends classical propagation-of-singularities results for Hamilton–Jacobi equations to the constrained Riemannian setting with an obstacle. The proofs rely on the distance function satisfying the eikonal equation away from the obstacle together with adapted propagation techniques; the explicit sharpness examples, including the nonconvex case with boundary differentiability, strengthen the contribution by demonstrating that the statements cannot be improved in general.

minor comments (3)
  1. [Introduction / Theorem statements] The statement of the main theorems would benefit from an explicit reference to the precise regularity assumed on the Riemannian metric (e.g., C^∞ or C^k) and on the obstacle (compactness alone or additional boundary regularity).
  2. Notation for the distance function d, the level sets {d = t}, and the singular set should be introduced once in a dedicated notation paragraph rather than piecemeal.
  3. [Examples section] In the sharpness examples, a brief indication of how the nonconvex obstacle is constructed (e.g., via a specific embedding or local chart) would help the reader verify the claimed differentiability at boundary points without consulting external references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of our results, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes its main results—that a compact obstacle forces singularities in sufficiently high level sets of the Riemannian distance function and that such singularities lie on nontrivial Lipschitz arcs—via direct analytic arguments based on the distance function satisfying the eikonal equation away from the obstacle together with standard propagation properties of Hamilton-Jacobi singularities. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the claims remain independent of the paper's own fitted quantities or prior outputs and are shown to be sharp by explicit counterexamples. This is the typical self-contained case for a pure existence/propagation theorem in analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background assumptions from Riemannian geometry and viscosity solution theory for Hamilton-Jacobi equations; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The Riemannian metric is smooth
    Invoked to ensure the distance function satisfies the eikonal-type Hamilton-Jacobi equation away from the obstacle and cut locus.
  • domain assumption The obstacle is compact
    Used to guarantee that the complement is open and that level sets of the distance function eventually lie entirely outside any fixed neighborhood of the obstacle.

pith-pipeline@v0.9.1-grok · 5614 in / 1270 out tokens · 26257 ms · 2026-06-25T23:49:48.869544+00:00 · methodology

discussion (0)

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Reference graph

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