Fractional Mean Curvature Flow

• Fractional Mean Curvature Flow is a generalization of classical curvature flow that replaces local curvature with a nonlocal, fractional curvature derived from the s-perimeter functional.
• It models interface evolution with rich dynamics and regularity properties, including volume-preserving variants that exhibit exponential convergence to spherical shapes.
• The theory leverages analytical and numerical methods, such as level-set formulations and variational schemes, to address singularity formation and stability challenges.

Fractional mean curvature flow refers to a family of geometric evolution equations that generalize classical mean curvature flow by replacing local curvature with a nonlocal, fractional counterpart. This nonlocality, parametrized by $s \in (0,1)$, yields rich dynamical and regularity phenomena unavailable to local flows. Fractional mean curvature flow arises naturally as the $L^2$-gradient flow of the $s$-perimeter functional, modeling interface evolution driven by nonlocal surface tension. The volume-preserving variant, central to recent work, constrains the total enclosed volume, leading to qualitative behavior distinct from unconstrained flows. The theory incorporates definitions, a priori estimates, regularity theory, singularity formation, long-time asymptotics, stability, and numerical and analytical methodologies, with connections to optimal transportation, probability, and nonlinear PDE.

1. Mathematical Formulation of Fractional Mean Curvature Flow

The $s$-fractional mean curvature of a set $E \subset \mathbb{R}^{n+1}$ at a boundary point $x \in \partial E$ is: $$H_s(x, E) := \mathrm{P.V.} \int_{\mathbb{R}^{n+1}} \left[\chi_{\mathbb{R}^{n+1} \setminus E}(y) - \chi_E(y)\right] |x-y|^{-(n+1+s)} dy$$ where $\chi$ denotes the characteristic function, and "P.V." is the principal value (Julin et al., 2023). This nonlocal curvature arises as the first variation of the $s$-perimeter functional: $$\operatorname{Per}_s(E) = \int_E \int_{E^c} |x-y|^{-(n+1+s)} dx\,dy$$ The basic evolution equation for the unconstrained flow is: $V_n(x,t) = -H_s(x, E_t)$ where $V_n$ is the outer normal velocity of $\partial E_t$ (Sáez et al., 2015).

For the volume-preserving variant, the flow is constrained as: $$V_n(x,t) = H_s(x, E_t) - \overline{H_s}(E_t) \qquad \overline{H_s}(E_t) = \frac{1}{|\partial E_t|} \int_{\partial E_t} H_s(x, E_t) d\mathcal{H}^n(x)$$ ensuring volume conservation throughout the evolution (Julin et al., 2023).

Level-set and viscosity solution formulations are used to handle non-smooth and singular evolutions: $$\partial_t u + H_s[x, u(\cdot, t)]\,|\nabla u| = 0,$$ where $u(\cdot, t)$'s zero-level set represents the evolving surface at time $t$ (Cinti et al., 2016).

2. Regularity Theory and A Priori Estimates

A foundational result for convex initial data is that a priori control can be established both on the geometry and the curvature. If $E_0$ is convex and smooth, then for all times in its interval of smooth existence:

• Convexity is preserved: $E_t$ remains convex.
• Inner and outer radii are uniformly bounded: $0 < r < R$ such that $B_r \subset E_t \subset B_R$ (up to translation).
• Fractional curvature is globally bounded: $\sup_{t<T} \sup_{\partial E_t} |H_s| \leq C$ (Cinti et al., 2018).

This enables two-step regularity boosting: first, convexity and bounded curvature quantitatively yield $C^{1+s}$ regularity for evolving hypersurfaces, via local graph representations and estimates of the support function. Explicitly, if locally $\partial E_t$ is given by the graph of $u$, boundedness of $H_s$ forces Hölder continuity: $|\nabla u(x) - \nabla u(y)| \leq C |x-y|^s$ yielding global $C^{1+s}$ estimates (Julin et al., 2023).

To upgrade to $C^{2+s+\alpha}$ regularity, the flow is parametrized by a radial height function $h(\cdot, t)$ on the sphere $S^n$; differentiating the evolution in tangent directions yields: $$\partial_t(\partial_X h) = \mathcal{L}[h](\partial_X h) + \text{lower-order terms}$$ with $\mathcal{L}[h]$ a linear integro-differential operator of order $1+s$. Sharp parabolic Schauder estimates for such operators, adapted to the geometry of $S^n$, provide control of $C^{2+s+\alpha}$ norms. The bootstrap argument ensures the avoidance of $C^{2}$ blow-up and propagates regularity uniformly in time [(Julin et al., 2023), 23].

Convexity is only used for the initial $C^{1+s}$ estimate; the upgrade to $C^{2+s+\alpha}$ relies purely on quasilinear PDE theory, suggesting potential extension to non-convex domains remaining embedded (Julin et al., 2023).

3. Singularity Formation, Comparison Principles, and Uniqueness

In the classical local mean curvature flow, convexity or positivity of the second fundamental form prevents neck-pinches or singularity formation before extinction. In fractional mean curvature flow, CSV [10] use a nonlocal barrier argument: convexity preservation and uniform $H_s$ lower bound ensure evolving sets remain one-sided with respect to their tangent hyperplanes. This excludes finite-time singularities, establishing long-time existence for convex flows (Cinti et al., 2018, Julin et al., 2023).

For arbitrary sets, the comparison principle holds for smooth solutions and viscosity solutions. If $E_t$ and $F_t$ are two such flows with $\overline{E}_0 \subset F_0^\circ$, then $E_t \subset F_t$ for all $t$ of joint existence. Uniqueness and finite-time extinction are consequences: any compact set shrinks to a point in finite time, unless constrained otherwise (Sáez et al., 2015).

Non-convex sets can develop singularities, including neck-pinches or fattening of level sets. Cinti–Sinestrari–Valdinoci demonstrate that, even in two dimensions (violating Grayson's classical theorem), a smooth embedded curve under fractional flow can split before vanishing, driven by nonlocal positive curvature in thin neck regions (Cinti et al., 2016). The fine classification of singularities in the nonlocal regime remains open.

4. Long-Time Behavior and Exponential Convergence

For convex sets or nearly-spherical initial data under volume-preserving flow, the only stationary states with constant fractional mean curvature are round balls (fractional Alexandrov theorem) [2,11]. Cesaroni–Novaga established that global $C^{2+s+\alpha}$ solutions converge exponentially fast to a ball of fixed volume: $$\| h(\cdot, t) - 1 \|_{C^k(S^n)} \leq C_k e^{-\lambda t} \qquad x_0(t) \to x_\ast \text{ exponentially}$$ for all $k \in \mathbb{N}$, with rates depending on initial $C^k$ norms (Cesaroni et al., 2022, Julin et al., 2023).

The underlying mechanism is strict energy dissipation: the $s$-perimeter is strictly decreasing except at balls, and a quantitative Poincaré-type inequality connects curvature oscillations to deviations from sphericity (Daniele et al., 2022). Lyapunov function arguments, spectral-gap estimates for the Riesz operator, and Grönwall-type arguments lead to exponential $C^k$ convergence. This holds both for volume-preserving flow and unconstrained evolution of periodic graphs.

Numerical time-discrete schemes for volume-preserving flow (variational minimization at each step) similarly yield exponential convergence to balls, with explicit discrete dissipation estimates (Daniele et al., 2022). In high regularity regimes or for nearly-spherical sets, rates can be quantitatively tracked via nonlocal Alexandrov-type refinements.

5. Analytical and Numerical Methods

Several approaches are available for analysis and computation of fractional mean curvature flows:

• Level-set/viscosity methods: Formulate the flow as a PDE for a function $u(x,t)$ whose superlevel sets evolve by fractional mean curvature; powerful comparison principles guarantee well-posedness even past singularities (Cinti et al., 2016, Cagnetti et al., 8 Apr 2025).
• Parametric PDEs for height functions: For convex or nearly-spherical sets, reparametrize $\partial E_t$ via normal or radial height; nonlocal quasilinear parabolic PDEs for the height benefit from Schauder regularity theory (Julin et al., 2019, Julin et al., 2023).
• Threshold dynamics/MBO schemes: For certain kernels (possibly anisotropic), discretize in time via convolution and thresholding; consistency with the viscosity solution can be proved as the time step vanishes (Chambolle et al., 2016).
• Minimizing-movement/variational flows: In the weak or distributional setting, time-discrete minimization against nonlocal perimeter plus proximity cost yields robust, stable flows. Uniqueness and comparison extend to the weak regime (Cagnetti et al., 8 Apr 2025, Daniele et al., 2022).

Regularity theory leverages sharp $C^{\alpha}$ and $C^{2+s+\alpha}$ parabolic estimates for fractional operators on manifolds, most notably via adaptations of Mikulevicius–Pragarauskas [23]. Analytical bootstrapping, exploiting lower-order error control, propagates smoothness uniformly in time. In graphical settings, parabolic Hölder theory for nonlocal difference operators ensures uniform bounds for Lipschitz initial data and upgrades to instantaneous smoothness given sufficient starting regularity (Cesaroni et al., 2021).

6. Extensions, Generalizations, and Open Problems

The techniques for regularity upgrade, a priori estimates, and exponential convergence admit generalization:

• Anisotropic kernels: The regularity arguments can be extended from isotropic to anisotropic fractional perimeters, treating nonlocal Wulff-type flows (Chambolle et al., 2016).
• Non-convex domains and avoidance principles: For non-convex sets, the main open problem remains the global description of singularity formation; convexity cannot be expected to persist, and a general comparison or avoidance principle analogous to the local case is absent (Cinti et al., 2016, Julin et al., 2023).
• Graphical and star-shaped regimes: Under graphical hypotheses, eventual regularization (emergence of global Lipschitz graphs) can occur over finite time intervals, driven by nonlocality—a property absent in the classical regime (Cameron, 2019, Cesaroni et al., 2021).
• Supercritical kernels ($s \geq 1$): Via core-radius cutoff and appropriate scaling, the behavior of supercritical fractional flows converges to the classical local mean curvature flow, both in perimeter and curvature, as $\varepsilon \to 0$ (Luca et al., 2021).
• Numerical analysis: Threshold-dynamics algorithms and time-discrete variational schemes provide robust computational tools; their consistency, stability, and convergence to viscosity or weak solutions are established in several works (Daniele et al., 2022, Chambolle et al., 2016, Cagnetti et al., 8 Apr 2025).

Broader applications arise in phase-transition models, optimal transport, interface dynamics in nonlocal media, and dislocation theory via anisotropic variants. The classification and dynamics of singularities, sharp rates of convergence, and quantitative stability under almost-constant curvature remain vibrant areas of ongoing research (Julin et al., 2023, Cesaroni et al., 2022).