https://neckpinch.com/authors/jingeon-an-lacroix/New arXiv papers by Jingeon An-Lacroix, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2512.14437/arxiv:2512.14437Tue, 16 Dec 2025 00:00:00 GMTJingeon An-Lacroix, Kiichi Tashiro — It is known that there is a strong relation between the parabolic Allen–Cahn equation and the mean curvature flow, in the sense that the parabolic Allen–Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow v=−H-∂_νlog |∇u|+f(u)/|∇u|, satisfied by level surfaces of any solution to the nonlinear parabolic equation ∂ₜu=Δu-f(u). Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen–Cahn equation \{beginalignedat2 ∂ₜu&=Δu &&in \|u|<1\ |∇u|&=1/ε &&on ∂\|u|<1\, endalignedat and confirm that under reasonable assumptions, the C^α norm of the forcing term ∂_νlog|∇u| converges to zero at an algebraic rate as ε→ 0, uniformly in time. This implies that the parabolic free boundary Allen–Cahn equation converges to the mean curvature flow, uniformly (in ε and in time) in the C^(2,α) sense.Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryPhase fields & multiphase flows https://neckpinch.com/papers/2507.22183/arxiv:2507.22183Tue, 29 Jul 2025 00:00:00 GMTJingeon An-Lacroix, Hardy Chan, Pak Tung Ho — Since the seminal paper of Graham and Zworski (Invent. Math. 2003), conformal geometric problems are studied in the fractional setting. We consider the convergence of fractional Yamabe flow, which is previously known under small initial energy assumption. Inspired by the deep work of Brendle (J. Diff. Geom. 2005), we obtain the full convergence result for arbitrary initial energy, whenever the (fractional) positive mass conjecture is valid.Yamabe flowPhysics