ACF-Monotonicity Formula on RCD(0,N) Metric Measure Cones

TL;DR

This work extends the Alt–Caffarelli–Friedman (ACF) monotonicity formula to non-smooth $RCD(0,N)$ metric measure cones $(C(\Sigma),d_C,m_C)$. The main result proves that, for nonnegative $u_1,u_2$ with $u_i(p)=0$, $u_1u_2=0$, and $\boldsymbol{\Delta} u_i \ge 0$, the functional $J(r)=\frac{1}{r^{4}}A_1(r)A_2(r)$, where $A_i(r)=\int_{B_r(p)} \frac{|\nabla u_i|^2}{d(x,p)^{N-2}}\,dm_C$, is monotone nondecreasing in $r>0$. The proof hinges on two ingredients: Stokes formulas on $RCD(K,N)$ spaces and a Polya–Szegő-type eigenvalue control (via Friedland–Hayman and Mondino–Semola), enabling comparison with eigenvalues on spherical caps. As an application, the paper derives a rigidity result: if $0< J(r_1)=J(r_2) < \infty$, then $(\Sigma,d_{\Sigma},m_{\Sigma})$ is a spherical suspension (and, in the smooth case, isometric to the sphere with explicit linear profiles for $u_1,u_2$). This work extends two-phase free boundary methods to non-smooth geometric settings and paves the way for further geometric and regularity results on $RCD$ spaces.

Abstract

The ACF-monotonicity formula is a powerful tool in the study of two-phase free boundary problems, which was introduced by Alt, Caffarelli, and Friedman[1]. In this paper, we extend it to RCD(0,N) metric measure cones. As an application, we give a rigidity result for RCD(0,N) metric measure cones.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.1
• Definition 2.3
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1