Nonlocal Ordered Mean Curvature with Integrable Kernel

TL;DR

The paper extends the classical ordered curvature symmetry results to a nonlocal setting with an integrable kernel $J\in L^1$ that is radially symmetric, compactly supported, and nonincreasing. It defines the nonlocal mean curvature $H^J_\Omega$ and proves a nonlocal analogue of Li–Nirenberg type symmetry: if $\Omega$ is open, bounded, connected, and satisfies monotone curvature, zero boundary measure, and interval-projection conditions, then $\Omega$ is symmetric about some hyperplane $\{x_n=\lambda_0\}$. The authors first establish that ordered curvature implies pairwise equality of curvature values along vertical fibers, using a localized perimeter analysis and a translation argument. They then generalize Alexandrov’s moving plane method to the nonlocal framework, showing that, either via an interior touching point or a nontransversal contact, symmetry about the critical plane follows. This work broadens nonlocal geometric analysis to rough domains and clarifies the role of integrable kernels in symmetry phenomena.

Abstract

In this paper we introduce and study the concept of nonlocal ordered curvature. In the classical (differential) setting, the problem was introduced by Nirenberg and Li, where they conjectured that if a bounded, smooth surface has its mean curvature ordered in a particular direction, then the surface must be symmetric with respect to some hyperplane orthogonal to that direction. The conjecture was proved by Li et al in 2022. Here we study the counterpart problem in the nonlocal setting, where the nonlocal mean curvature of a set $Ω$, at any point $x$ on its boundary, is defined as $H_Ω^J(x) = \int_{Ω^c} J(x-y) dy - \int_ΩJ(x-y) dy$ and the kernel function $J$ is radially symmetric, non-increasing, integrable and compactly supported. Using a generalization of Alexandrov's moving plane method, we prove a similar result in the nonlocal setting.

Figures (5)

• Figure 1: Types of boundary points (P1 - P4). The nonlocal curvature, $H^J_\Omega(x)$, is the difference between the $J$-weighted "outer" volume (red) and its $J$-weighted "inner" volume (green).
• Figure 2: The nonlocal curvature is ordered in the $x_n$-direction: $[a^1,b^1],[a^2,b^2]\subseteq\pi^{\text{inv}}_{\overline{\Omega}}(\widehat{x})$, so $H^J_\Omega(A^1)\le H^J_\Omega(B^1)$ and $H_\Omega^J(A^2)\le H_\Omega^J(B^2)$.
• Figure 3: Translation domains and relative complement components
• Figure 4: Identification of $z$ and $\widehat{z}$ in Case 1 for the proof of Lemma \ref{['lem:3lambda']}
• Figure 5: (Left) $k$-th step in proof of Lemma \ref{['lem:non_trans']}. (Right) The fixed cylinder $Q(x_0)\subseteq \Omega\cap\text{HP}_0^+\setminus R_0(\overline{\Omega})$, with $x_0\in\partial B_r(0)$.

Theorems & Definitions (26)

• Remark 1.5
• Theorem 1.6
• Theorem 2.1
• Theorem 2.2
• proof : Proof
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3