Fractional Mean Curvature

• Fractional mean curvature is a nonlocal analogue of classical mean curvature, defined via a principal-value singular integral that accounts for the global geometry of a set.
• It originates from the first variation of fractional perimeter functionals and provides the Euler–Lagrange framework for hypersurfaces with constant nonlocal curvature.
• This concept underpins nonlocal geometric flows and anisotropic models, converging to traditional curvature in the local limit and supporting stability and regularity analyses.

Fractional mean curvature is the nonlocal analogue of classical mean curvature associated with the first variation of a fractional perimeter. For a sufficiently regular set $E\subset \mathbb R^n$ and a boundary point $x\in\partial E$, it is defined through a principal-value singular integral of the jump of $\chi_E$ across $\partial E$, so its value depends on the whole geometry of $E$ rather than only on local second-order data. This quantity enters Euler–Lagrange equations for critical points of fractional perimeters, generates nonlocal geometric flows, admits anisotropic and fixed-boundary variants, and converges to classical mean curvature after the standard renormalization as the fractional order approaches $1$ (Sáez et al., 2015, Cesaroni et al., 2020).

1. Definitions and sign conventions

A standard isotropic definition for the $s$-fractional mean curvature, with $s\in(0,1)$, is

$$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$

possibly multiplied by a normalization factor such as $C_{n,s}$ or $x\in\partial E$0, depending on the paper. Equivalent forms include a surface integral over $x\in\partial E$1, obtained by the divergence theorem, for instance

$x\in\partial E$2

again up to normalization and sign. The principal value is essential because the kernel is singular at $x\in\partial E$3. Several papers emphasize that sign conventions differ: some write $x\in\partial E$4, others $x\in\partial E$5, and some choose the sign so that convex sets have nonnegative curvature (Cabre et al., 2016, Sáez et al., 2015, Julin et al., 2023).

A related surface formula, used in the study of fractional Willmore-type energies, writes the curvature at $x\in\partial E$6 as

$x\in\partial E$7

with $x\in\partial E$8 the exterior unit normal, and with $x\in\partial E$9 chosen so that $\chi_E$0, the classical mean curvature (Blatt et al., 2023). For hypersurfaces with boundary, Onoue uses a more general definition in terms of measurable interior and exterior regions $\chi_E$1 and $\chi_E$2 attached to a point $\chi_E$3, namely

$\chi_E$4

which recovers the usual indicator-function formula when $\chi_E$5 is a closed smooth hypersurface (Onoue, 2023).

An anisotropic analogue replaces the Euclidean kernel by a convex, even, one-homogeneous norm $\chi_E$6 satisfying $\chi_E$7. In that setting Chambolle, Novaga, and Ruffini define

$\chi_E$8

or equivalently $\chi_E$9, with the convention that convex sets have $\partial E$0 (Chambolle et al., 2016).

2. Variational origin and associated energies

Fractional mean curvature arises as the first variation of fractional perimeter functionals. In the isotropic case one writes, up to normalization,

$\partial E$1

or equivalently

$\partial E$2

If $\partial E$3 is a deformation of $\partial E$4 with normal speed $\partial E$5, then the first variation is

$\partial E$6

again with the appropriate convention for the sign. Under a volume constraint, criticality is therefore equivalent to constancy of the nonlocal mean curvature on the boundary: $\partial E$7 This is the Euler–Lagrange condition used in the construction of nonlocal Delaunay hypersurfaces and multiply periodic constant-nonlocal-mean-curvature hypersurfaces (Cabre et al., 2016, Minlend et al., 2018, Cinti et al., 2018).

For compact manifolds with fixed boundary, the first variation contains an interior term involving $\partial E$8 and a boundary term on $\partial E$9. When variations vanish on $E$0, the boundary term disappears, and stationarity is equivalent to

$E$1

This is the Euler–Lagrange equation for the fractional area functional introduced by Paroni, Podio-Guidugli, and Seguin and analyzed by Onoue (Onoue, 2023).

A higher-order functional built from fractional mean curvature is the fractional Willmore-type energy

$E$2

Under the dilation $E$3, this scales as

$E$4

so the critical exponent is $E$5, and the subcritical regime is $E$6. In the convex setting, the paper on this functional identifies $E$7 with a nonlocal bending energy $E$8 and derives local graph control, lower Ahlfors-regularity, a weak Michael–Simon inequality, and a stability statement toward spheres (Blatt et al., 2023).

3. Fractional mean curvature flows

The basic geometric evolution prescribes that the normal velocity equals minus the fractional mean curvature: $E$9 In star-shaped and graphical parametrizations this becomes, respectively,

$1$0

and

$1$1

For smooth solutions Sáez and Valdinoci established a comparison principle, uniqueness, finite-time extinction for compact data, the evolution formula for the fractional perimeter,

$1$2

and an evolution equation for $1$3 featuring a nonlocal diffusion term plus a nonnegative normal-difference term (Sáez et al., 2015).

In the volume-preserving variant one evolves by

$1$4

where

$1$5

This choice enforces $1$6. Julin and La Manna proved that if $1$7 is convex, $1$8, and has the same volume as the unit ball, then the classical solution exists for all $1$9 and converges exponentially fast to a translate of $s$0; after a time $s$1, one can write

$s$2

with

$s$3

Their proof upgrades a priori bounds to $s$4, then to $s$5, and concludes that no finite-time singularity occurs for convex data (Julin et al., 2023).

Weak and discrete formulations are central in the nonlocal setting. Chambolle, Novaga, and Ruffini introduced an anisotropic threshold-dynamics scheme based on

$s$6

and the one-step map

$s$7

As $s$8, the piecewise-constant approximation converges locally uniformly to the unique viscosity solution of the level-set PDE

$s$9

where

$s\in(0,1)$0

This gives a consistent threshold-dynamics approximation for anisotropic fractional mean curvature flow with a continuous time-dependent forcing term (Chambolle et al., 2016).

Graphical and boundary-value settings lead to further variants. For entire Lipschitz graphs, the graph equation is a quasilinear integro-differential parabolic equation of order $s\in(0,1)$1, with kernel bounds $s\in(0,1)$2; the corresponding level-set formulation admits a unique global viscosity solution for bounded uniformly continuous initial data, preserves Lipschitz constants, and smooths Lipschitz graphs to $s\in(0,1)$3 for any positive time (Cesaroni et al., 2021). If an initial open set lies between two parallel Lipschitz subgraphs, then after a universal time $s\in(0,1)$4 the minimal viscosity supersolution becomes exactly the subgraph of a $s\in(0,1)$5-Lipschitz function, with

$s\in(0,1)$6

a regularizing effect that the paper states is false for the classical mean curvature flow (Cameron, 2019). Short-time classical existence for bounded $s\in(0,1)$7 initial hypersurfaces, with the same result for the volume-preserving flow, was established by Julin and La Manna through a fractional Schauder fixed-point argument (Julin et al., 2019). A capillary version in the half-space, with constant contact angle $s\in(0,1)$8, was later formulated as

$s\in(0,1)$9

and reduced, in radial variables, to a nonlocal scalar PDE on $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$0 with boundary condition

$$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$1

short-time existence follows from a contraction argument (Fan et al., 8 Feb 2026).

4. Stationary hypersurfaces and constant nonlocal mean curvature

Critical points of fractional perimeter under a volume constraint are hypersurfaces with constant nonlocal mean curvature. Cabré, Fall, and Weth proved the existence of a smooth branch of periodic cylinders in $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$2, $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$3, all with the same constant nonlocal mean curvature and bifurcating from a straight cylinder. For $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$4, they obtain a family

$$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$5

with

$$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$6

where $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$7 is even and periodic, $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$8, $$H_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_{E^c}(y)-\chi_E(y)}{|x-y|^{n+s}}\,dy,$$9 in $C_{n,s}$0, and the nonlocal mean curvature equals $C_{n,s}$1. The proof uses the Crandall–Rabinowitz theorem applied to a quasilinear type fractional elliptic equation (Cabre et al., 2016).

A higher-dimensional periodic analogue was obtained for stacked slabs. Dávila, Del Pino, Dipierro, and Valdinoci construct smooth branches of multiply-periodic hypersurfaces of constant nonlocal mean curvature bifurcating from suitable parallel hyperplanes. Their Lyapunov–Schmidt reduction isolates a one-dimensional kernel in a symmetry-restricted space and produces nontrivial solutions of

$C_{n,s}$2

yielding genuinely non-flat $C_{n,s}$3-periodic hypersurfaces (Minlend et al., 2018).

The zero-curvature case with fixed boundary shows a different rigidity. Onoue proves that if the boundary is a single $C_{n,s}$4-sphere $C_{n,s}$5, then any orientable compact $C_{n,s}$6 manifold $C_{n,s}$7 with $C_{n,s}$8 and $C_{n,s}$9 must coincide with the flat disk

$x\in\partial E$00

For two parallel spherical boundaries $x\in\partial E$01, critical points do not coincide with the two horizontal caps, do not touch the vertical side-wall of the spanning cylinder, and exhibit a gap-dependent topology: for sufficiently large separation $x\in\partial E$02, any critical $x\in\partial E$03 splits into exactly two components $x\in\partial E$04 with $x\in\partial E$05; for sufficiently small gap and $x\in\partial E$06, the two boundaries belong to the same connected component (Onoue, 2023).

5. Convexity, regularity, and geometric inequalities

Convexity plays a central role in the regularity theory. In the anisotropic threshold scheme of Chambolle, Novaga, and Ruffini, each discrete step $x\in\partial E$07 preserves convexity because the kernel $x\in\partial E$08 is a $x\in\partial E$09-concave, integrable radial kernel. Passing to the limit, if all superlevel sets $x\in\partial E$10 are convex, then $x\in\partial E$11 remain convex for all $x\in\partial E$12. For bounded convex initial sets, the same paper proves that the geometric flow with normal velocity $x\in\partial E$13 does not fatten and defines a unique evolution, based on a distance-nondecreasing comparison in the normal direction (Chambolle et al., 2016).

For the volume-preserving flow of smooth convex sets, Cinti, Sinestrari, and Valdinoci show uniform control of the inner and outer radii and a two-sided curvature bound

$x\in\partial E$14

Under a continuation criterion requiring a $x\in\partial E$15 bound from a uniform $x\in\partial E$16 bound, they obtain long-time existence and convergence, up to translation, to a sphere in $x\in\partial E$17. Julin and La Manna later proved that for convex $x\in\partial E$18 initial data the flow does not develop singularities at all and converges exponentially fast to a ball, with the regularity step from $x\in\partial E$19 to $x\in\partial E$20 not relying on convexity (Cinti et al., 2018, Julin et al., 2023).

For entire Lipschitz graphs, a different regularization mechanism is available. The graph flow has an elliptic nonlocal operator of order $x\in\partial E$21, and viscosity solutions become $x\in\partial E$22 for every positive time. If the initial graph is a sublinear perturbation of a cone, then in a rescaled framework the evolution converges in $x\in\partial E$23 to the unique expanding self-similar profile associated to the cone. Hyperplanes are stable, convex cones have homothetic expansions, and any uniformly Lipschitz ancient graphical solution is stationary; in particular, any homothetic shrinking entire Lipschitz graph must be a hyperplane (Cesaroni et al., 2021).

Fractional mean curvature also governs Sobolev and geometric inequalities on convex hypersurfaces. Cabré, Cozzi, and Csató proved a fractional Michael–Simon–Allard inequality: if $x\in\partial E$24 is convex, $x\in\partial E$25, and $x\in\partial E$26, then

$x\in\partial E$27

As an application, for convex smooth fractional mean curvature flow they derive an upper bound for the maximal existence time,

$x\in\partial E$28

depending on the perimeter of the initial set rather than its diameter (Cabre et al., 2020). In the subcritical fractional Willmore setting, if $x\in\partial E$29 is the smooth boundary of a compact convex body, $x\in\partial E$30, and $x\in\partial E$31, then for every $x\in\partial E$32 there is a graph parametrization $x\in\partial E$33 on a ball $x\in\partial E$34 with

$x\in\partial E$35

The paper deduces uniform lower Ahlfors-regularity, a weak Michael–Simon inequality, and a stability theorem asserting Hausdorff closeness to a round sphere under smallness of the support-function deviation in $x\in\partial E$36 (Blatt et al., 2023).

6. Limit regimes, supercritical regularizations, and phase-field derivations

A systematic limit theory clarifies how fractional mean curvature interpolates between nonlocal and local geometries. Cesàroni, De Luca, Novaga, and Ponsiglione prove that for smooth $x\in\partial E$37,

$x\in\partial E$38

where $x\in\partial E$39 is the classical mean curvature. As $x\in\partial E$40,

$x\in\partial E$41

a geometry-independent constant, and after subtraction of $x\in\partial E$42 one obtains a nontrivial zero-order curvature $x\in\partial E$43. The associated level-set flows converge after the corresponding time rescalings: as $x\in\partial E$44, the rescaled fractional flow converges to classical mean curvature flow; as $x\in\partial E$45, the rescaled flow converges first to constant-speed motion and, after subtraction of the leading term, to the flow driven by $x\in\partial E$46 (Cesaroni et al., 2020).

A different local limit comes from supercritical kernels. De Luca, Kubin, and Ponsiglione introduce the core-radius regularized kernel

$x\in\partial E$47

and the regularized curvature

$x\in\partial E$48

With

$x\in\partial E$49

they show

$x\in\partial E$50

and the corresponding nonlocal level-set flows converge, after the time change $x\in\partial E$51, to classical mean-curvature flow. The same construction extends to anisotropic kernels $x\in\partial E$52, recovering anisotropic mean curvature in the limit and including the line-tension energy of planar dislocations as a special case (Luca et al., 2021).

Fractional mean curvature also appears as the sharp interface limit of nonlocal phase-field models. In the derivation based on fractional Laplacians of order $x\in\partial E$53, one compares the $x\in\partial E$54-dimensional operator applied to the layer ansatz $x\in\partial E$55 with the one-dimensional operator satisfied by the profile $x\in\partial E$56, and defines a difference quantity $x\in\partial E$57. The key convergence theorem states that $x\in\partial E$58, where $x\in\partial E$59 is the fractional mean curvature of the interface. In this normalization the curvature is written

$x\in\partial E$60

reflecting the fact that the nonlocal diffusion is parameterized by the Laplacian order $x\in\partial E$61. The same analysis shows that as $x\in\partial E$62 one recovers the classical identity $x\in\partial E$63, and in the fractional Allen–Cahn evolution the moving interface satisfies

$x\in\partial E$64

yielding nonlocal curvature-driven motion for $x\in\partial E$65 and classical mean-curvature motion in the local limit (Patrizi et al., 2024).