Nonlocal Ordered Mean Curvature with Non-Integrable Kernel

Active research. This page is intentionally a problem statement rather than a progress report. Current lemmas, conjectures, calculations, experiments, and proof strategies are omitted while the project is active.

Background

Classical curvature-rigidity problems ask when curvature information forces a hypersurface to be symmetric. Alexandrov's moving-plane method is one of the standard tools. Nonlocal curvature replaces a pointwise differential quantity by an integral interaction between a set and its complement, weighted by a kernel.

The singular, non-integrable regime changes the analysis near the boundary and makes reflection arguments more delicate than in the integrable-kernel setting.

Problem

Understand when an ordering condition on nonlocal mean curvature is strong enough to force geometric rigidity or symmetry for singular, non-integrable interaction kernels, and determine which hypotheses are genuinely structural.

Tools

The moving-plane problem needs a few standard geometric and analytic objects fixed first. These are background definitions and classical results, not the active comparison argument.

Nonlocal Mean Curvature

Definition. Let \(\Omega\subset\mathbb R^n\) and let \(J:\mathbb R^n\setminus\{0\}\to[0,\infty)\). At a boundary point \(x\), define formally \[ H_\Omega^J(x) = \operatorname{PV}\int_{\mathbb R^n} \big(\chi_{\Omega^c}(y)-\chi_\Omega(y)\big)J(x-y)\,dy, \] whenever the principal value exists.

Radial Kernels

Definition. The kernel \(J\) is radial and nonincreasing if \(J(z)=\mu(|z|)\) for some \(\mu\) with \(\mu(r_1)\geq\mu(r_2)\) whenever \(r_1

In the singular regime one may have \(J\notin L^1\) near the origin, so cancellation and boundary regularity become part of the analysis.

Reflection

Definition. For a unit vector \(e\) and \(\lambda\in\mathbb R\), let \[ T_\lambda=\{x:x\cdot e=\lambda\}. \] Reflection across \(T_\lambda\) is \[ x^\lambda=x-2(x\cdot e-\lambda)e. \]

Classical Moving Planes

Theorem (Alexandrov). A compact connected embedded \(C^2\) hypersurface in Euclidean space with constant mean curvature is a sphere.

The methodological point is the reflection argument: move a plane until first contact and compare the reflected and original surfaces. The active singular-kernel contact analysis is intentionally omitted.

Direction

The active work is about locating the boundary between assumptions that are genuinely necessary and assumptions inherited from earlier formulations. The current comparison identities, case decompositions, and proof architecture are intentionally not public here.

Last updated: September 14, 2026.