---
title: Fractional Mean Curvature
url: https://www.emergentmind.com/topics/fractional-mean-curvature
type: topic
---

# Fractional Mean Curvature

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 \(1\) [1511.06944, 2003.02248].

## 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 \(s(1-s)\), depending on the paper. Equivalent forms include a surface integral over \(\partial E\), obtained by the divergence theorem, for instance
\[
H_s(x)=-\int_{\partial E}\frac{(x-y)\cdot \nu_E(y)}{|x-y|^{N+s}}\,d\sigma(y),
\]
again up to normalization and sign. The principal value is essential because the kernel is singular at \(y=x\). Several papers emphasize that sign conventions differ: some write \(\chi_{E^c}-\chi_E\), others \(\chi_E-\chi_{E^c}\), and some choose the sign so that convex sets have nonnegative curvature [1602.02623, 1511.06944, 2307.03912].

A related surface formula, used in the study of fractional Willmore-type energies, writes the curvature at \(x\in\Sigma=\partial E\) as
\[
H^s(x)=C_{n,s}\int_{\Sigma}\frac{\langle x-y,n(y)\rangle}{|x-y|^{n+s}}\,d\mathcal H^{n-1}(y),
\]
with \(n(y)\) the exterior unit normal, and with \(C_{n,s}\) chosen so that \(\lim_{s\to1^-}H^s(x)=H(x)\), the classical mean curvature [2306.16941]. For hypersurfaces with boundary, Onoue uses a more general definition in terms of measurable interior and exterior regions \(A_i(z)\) and \(A_e(z)\) attached to a point \(z\in M\), namely
\[
H_{M,s}(z)=c_{N,s}\,\mathrm{PV}\!\int_{\mathbb R^N}\frac{\chi_{A_i(z)}(y)-\chi_{A_e(z)}(y)}{|y-z|^{N+s}}\,dy,
\]
which recovers the usual indicator-function formula when \(M=\partial E\) is a closed smooth hypersurface [2310.11567].

An anisotropic analogue replaces the Euclidean kernel by a convex, even, one-homogeneous norm \(N\colon\mathbb R^n\to[0,\infty)\) satisfying \(c|x|\le N(x)\le C|x|\). In that setting Chambolle, Novaga, and Ruffini define
\[
K_s(x,E)=\mathrm{PV}\!\int_{\mathbb R^n}\bigl[\chi_{E^c}(y)-\chi_E(y)\bigr]\,N(y-x)^{-(n+s)}\,dy,
\]
or equivalently \(-K_s(x,E)=\int (\chi_E-\chi_{E^c})N(y-x)^{-(n+s)}\,dy\), with the convention that convex sets have \(K_s\ge0\) [1603.07239].

## 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,
\[
\mathrm{Per}_s(E)=\int_E\int_{E^c}\frac{dx\,dy}{|x-y|^{n+s}},
\]
or equivalently
\[
\mathrm{Per}_s(E)=\frac12\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|\chi_E(x)-\chi_E(y)|}{|x-y|^{n+s}}\,dx\,dy.
\]
If \(E_t\) is a deformation of \(E\) with normal speed \(V\), then the first variation is
\[
\frac{d}{dt}\mathrm{Per}_s(E_t)\Big|_{t=0}=\int_{\partial E}H_s(x)\,V(x)\,d\sigma(x),
\]
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:
\[
H_s(x)=\lambda,\qquad x\in\partial E.
\]
This is the Euler–Lagrange condition used in the construction of nonlocal Delaunay hypersurfaces and multiply periodic constant-nonlocal-mean-curvature hypersurfaces [1602.02623, 1804.01782, 1811.08651].

For compact manifolds with fixed boundary, the first variation contains an interior term involving \(H_{M,s}\) and a boundary term on \(\partial M\). When variations vanish on \(\partial M\), the boundary term disappears, and stationarity is equivalent to
\[
H_{M,s}(z)=0,\qquad z\in M.
\]
This is the Euler–Lagrange equation for the fractional area functional introduced by Paroni, Podio-Guidugli, and Seguin and analyzed by Onoue [2310.11567].

A higher-order functional built from fractional mean curvature is the fractional Willmore-type energy
\[
E_{s,p}(\Sigma)=\int_\Sigma |H^s(x)|^p\,d\mathcal H^{n-1}(x).
\]
Under the dilation \(x\mapsto \lambda x\), this scales as
\[
E_{s,p}(\lambda\Sigma)=\lambda^{\,n-1-sp}E_{s,p}(\Sigma),
\]
so the critical exponent is \(p=(n-1)/s\), and the subcritical regime is \(p>(n-1)/s\). In the convex setting, the paper on this functional identifies \(E_{s,p}\) with a nonlocal bending energy \(B_{s,p}\) and derives local graph control, lower Ahlfors-regularity, a weak Michael–Simon inequality, and a stability statement toward spheres [2306.16941].

## 3. Fractional mean curvature flows

The basic geometric evolution prescribes that the normal velocity equals minus the fractional mean curvature:
\[
\partial_t x(t)\cdot \nu=-H_s\bigl(x(t),E_t\bigr),\qquad x(t)\in\partial E_t.
\]
In star-shaped and graphical parametrizations this becomes, respectively,
\[
\partial_t f(\omega,t)=-H_s(f(\omega,t)\omega,E_t)\,\sqrt{|\nabla_{S^{n-1}}f|^2+f^2}
\]
and
\[
\partial_t u(x',t)=-\sqrt{1+|\nabla_{x'}u|^2}\,H_s\bigl((x',u(x',t)),E_t\bigr).
\]
For smooth solutions Sáez and Valdinoci established a comparison principle, uniqueness, finite-time extinction for compact data, the evolution formula for the fractional perimeter,
\[
\frac{d}{dt}P_s(E_t)=-\int_{\partial E_t}H_s(x,E_t)^2\,d\mathcal H^{n-1}(x)\le0,
\]
and an evolution equation for \(H_s\) featuring a nonlocal diffusion term plus a nonnegative normal-difference term [1511.06944].

In the volume-preserving variant one evolves by
\[
\partial_tX(p,t)=\bigl(-H_s(\partial E_t,X(p,t))+\lambda(t)\bigr)\nu(p,t),
\]
where
\[
\lambda(t)=\frac{\int_{\partial E_t}H_s(y,t)\,d\mathcal H^n(y)}{\int_{\partial E_t}d\mathcal H^n}
=\avg_{\partial E_t}H_s(\cdot,t).
\]
This choice enforces \(\frac{d}{dt}|E_t|=0\). Julin and La Manna proved that if \(E_0\subset\mathbb R^{n+1}\) is convex, \(C^{1,1}\), and has the same volume as the unit ball, then the classical solution exists for all \(t\ge0\) and converges exponentially fast to a translate of \(B_1\); after a time \(T\), one can write
\[
\partial E_t=\{\,h(x,t)x+x_0:\ x\in S^n\},
\]
with
\[
\|h(\cdot,t)-1\|_{C^k(S^n)}\le C_k e^{-\alpha t}.
\]
Their proof upgrades a priori bounds to \(C^{1+s}\), then to \(C^{2+\alpha}\), and concludes that no finite-time singularity occurs for convex data [2307.03912].

Weak and discrete formulations are central in the nonlocal setting. Chambolle, Novaga, and Ruffini introduced an anisotropic threshold-dynamics scheme based on
\[
P(x)=\frac1{1+N(x)^{n+s}},\qquad
P_h(x)=h^{-n/(n+s)}P\!\left(\frac{x}{h^{1/(n+s)}}\right),
\]
and the one-step map
\[
T_{g(t),h}(E)=\{x\in\mathbb R^n:\ P_h*(\chi_E-\chi_{E^c})(x)>g(t)\,h\}.
\]
As \(h\to0\), the piecewise-constant approximation converges locally uniformly to the unique viscosity solution of the level-set PDE
\[
(\partial_tu)(x,t)=A(Du)|Du|\,[ -K_s(x,\{u\ge u(x,t)\})+g(t)],
\]
where
\[
A(p)=|p|^{-1}\int_{p^\perp}P(y)\,d\mathcal H^{n-1}(y).
\]
This gives a consistent threshold-dynamics approximation for anisotropic fractional mean curvature flow with a continuous time-dependent forcing term [1603.07239].

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 \(1+s\), with kernel bounds \(\lambda\le A\le \Lambda\); 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 \(C^{1+\alpha}\) for any positive time [2103.11346]. If an initial open set lies between two parallel Lipschitz subgraphs, then after a universal time \(t\ge R^{1+s}T(d,s,L)\) the minimal viscosity supersolution becomes exactly the subgraph of a \((1+L)\)-Lipschitz function, with
\[
T(d,s,L)\le C(d)\,\frac{(1+L)^{d+s}}{s^2(1-s)},
\]
a regularizing effect that the paper states is false for the classical mean curvature flow [1905.09184]. Short-time classical existence for bounded \(C^{1,1}\) 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 [1906.10990]. A capillary version in the half-space, with constant contact angle \(\theta\in(0,\pi)\), was later formulated as
\[
(\partial_t\iota)^\perp=-H^s\nu,\qquad
\langle \nu,\bar N\circ\iota\rangle=-\cos\theta,
\]
and reduced, in radial variables, to a nonlocal scalar PDE on \(S^n_+\) with boundary condition
\[
\partial_\eta\rho=\cos\theta\,\sqrt{\rho^2+|\nabla_\tau\rho|^2};
\]
short-time existence follows from a contraction argument [2602.07989].

## 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 \(\mathbb R^N\), \(N\ge2\), all with the same constant nonlocal mean curvature and bifurcating from a straight cylinder. For \(a\in(0,1)\), they obtain a family
\[
E_{u_\varepsilon}=\{(s,x')\in\mathbb R\times\mathbb R^{N-1}: |x'|<u_\varepsilon(s)\},
\]
with
\[
u_\varepsilon(s)=R+\frac1{X(\varepsilon)}\bigl[\cos(X(\varepsilon)s)+v_\varepsilon(X(\varepsilon)s)\bigr],
\]
where \(u_\varepsilon\) is even and periodic, \(u_0\equiv R\), \(v_\varepsilon\to0\) in \(C^{1,\beta}\), and the nonlocal mean curvature equals \(H_a(R)\). The proof uses the Crandall–Rabinowitz theorem applied to a quasilinear type fractional elliptic equation [1602.02623].

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
\[
H(T,R+y)=H(T,R),
\]
yielding genuinely non-flat \(2T\mathbb Z^{n-1}\times\mathbb Z\)-periodic hypersurfaces [1804.01782].

The zero-curvature case with fixed boundary shows a different rigidity. Onoue proves that if the boundary is a single \((N-2)\)-sphere \(\Gamma_0\subset\{x_N=0\}\), then any orientable compact \(C^{1,\alpha}\) manifold \(M\) with \(\partial M=\Gamma_0\) and \(H_{M,s}\equiv0\) must coincide with the flat disk
\[
M=\overline G\times\{0\}.
\]
For two parallel spherical boundaries \(\Gamma_1\cup\Gamma_2\), 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 \(d\), any critical \(M\) splits into exactly two components \(M_1,M_2\) with \(\partial M_i=\Gamma_i\); for sufficiently small gap and \(N\ge3\), the two boundaries belong to the same connected component [2310.11567].

## 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 \(T_{g(nh),h}\) preserves convexity because the kernel \(P_h\) is a \(( -1/(n+s))\)-concave, integrable radial kernel. Passing to the limit, if all superlevel sets \(\{u_0>s\}\) are convex, then \(\{u(\cdot,t)>s\}\) remain convex for all \(t>0\). For bounded convex initial sets, the same paper proves that the geometric flow with normal velocity \(V=-K_s+g(t)\) does not fatten and defines a unique evolution, based on a distance-nondecreasing comparison in the normal direction [1603.07239].

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
\[
0<K_1\le H_s(x,t)\le K_2<\infty.
\]
Under a continuation criterion requiring a \(C^{2,\beta}\) bound from a uniform \(H_s\) bound, they obtain long-time existence and convergence, up to translation, to a sphere in \(C^{2,\alpha}\). Julin and La Manna later proved that for convex \(C^{1,1}\) initial data the flow does not develop singularities at all and converges exponentially fast to a ball, with the regularity step from \(C^{1+\alpha}\) to \(C^{2+\alpha}\) not relying on convexity [1811.08651, 2307.03912].

For entire Lipschitz graphs, a different regularization mechanism is available. The graph flow has an elliptic nonlocal operator of order \(1+s\), and viscosity solutions become \(C^{1+\alpha}\) for every positive time. If the initial graph is a sublinear perturbation of a cone, then in a rescaled framework the evolution converges in \(C^1_{\mathrm{loc}}\) 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 [2103.11346].

Fractional mean curvature also governs Sobolev and geometric inequalities on convex hypersurfaces. Cabré, Cozzi, and Csató proved a fractional Michael–Simon–Allard inequality: if \(M=\partial E\) is convex, \(n>sp\), and \(p_s=np/(n-sp)\), then
\[
\left(\int_M |u|^{p_s}\,dS\right)^{1/p_s}
\le C(n,s,p)\left([u]_{W^{s,p}(M)}^p+\int_M H_s(x)|u(x)|^p\,dS(x)\right)^{1/p}.
\]
As an application, for convex smooth fractional mean curvature flow they derive an upper bound for the maximal existence time,
\[
T^*\le C(n,s)\,|\partial E_0|^{(1+s)/n},
\]
depending on the perimeter of the initial set rather than its diameter [2004.13129]. In the subcritical fractional Willmore setting, if \(\Sigma=\partial E\) is the smooth boundary of a compact convex body, \(p>(n-1)/s\), and \(E_{s,p}(\Sigma)\le A\), then for every \(x_0\in\Sigma\) there is a graph parametrization \(\Phi(x)=x_0+P(x,f(x))\) on a ball \(B(0,R)\) with
\[
[f]_{C^{\,s-(n-1)/p}(B(0,R))}\le C.
\]
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 \(W^{a,q}\) [2306.16941].

## 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 \(E\subset\mathbb R^d\),
\[
\lim_{s\to1^-}(1-s)H^s(x;E)=d(d-1)\omega_{d-2}\,H^1(x;E),
\]
where \(H^1\) is the classical mean curvature. As \(s\to0^+\),
\[
\lim_{s\to0^+} s\,H^s(x;E)=d\,\omega_d,
\]
a geometry-independent constant, and after subtraction of \(d\omega_d/s\) one obtains a nontrivial zero-order curvature \(H^0\). The associated level-set flows converge after the corresponding time rescalings: as \(s\to1^-\), the rescaled fractional flow converges to classical mean curvature flow; as \(s\to0^+\), the rescaled flow converges first to constant-speed motion and, after subtraction of the leading term, to the flow driven by \(H^0\) [2003.02248].

A different local limit comes from supercritical kernels. De Luca, Kubin, and Ponsiglione introduce the core-radius regularized kernel
\[
k_{s,\delta}(r)=
\begin{cases}
r^{-n-s}, & r\ge\delta,\\
\delta^{-n-s}, & 0<r<\delta,
\end{cases}
\]
and the regularized curvature
\[
H_{s,\delta}(x)=\int_{\mathbb R^n}\bigl(\chi_{E^c}(y)-\chi_E(y)\bigr)\,k_{s,\delta}(|x-y|)\,dy.
\]
With
\[
\alpha_s(\delta)=
\begin{cases}
\dfrac{n+s}{\delta^{s-1}}, & s>1,\\[4pt]
|\ln\delta|, & s=1,
\end{cases}
\]
they show
\[
\frac1{\alpha_s(\delta)}\,P_{s,\delta}\ \Gamma\text{-converges to }\ \omega_{n-1}\,\mathrm{Per}(E),
\qquad
\lim_{\delta\to0}\frac1{\alpha_s(\delta)}H_{s,\delta}(x)=\omega_{n-1}H(x),
\]
and the corresponding nonlocal level-set flows converge, after the time change \(\tau=\alpha_s(\delta)t\), to classical mean-curvature flow. The same construction extends to anisotropic kernels \(g(\frac{x-y}{|x-y|})k_{s,\delta}(|x-y|)\), recovering anisotropic mean curvature in the limit and including the line-tension energy of planar dislocations as a special case [2102.13041].

Fractional mean curvature also appears as the sharp interface limit of nonlocal phase-field models. In the derivation based on fractional Laplacians of order \(2s\), one compares the \(n\)-dimensional operator applied to the layer ansatz \(u^\varepsilon(x)=\phi(d(x)/\varepsilon)\) with the one-dimensional operator satisfied by the profile \(\phi\), and defines a difference quantity \(d^\varepsilon(x)\). The key convergence theorem states that \(d^\varepsilon(x)\to K[x,d]\), where \(K[x,d]\) is the fractional mean curvature of the interface. In this normalization the curvature is written
\[
H_s(x)=\mathrm{PV}\!\int_{\mathbb R^n}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x-y|^{n+2s}}\,dy,
\]
reflecting the fact that the nonlocal diffusion is parameterized by the Laplacian order \(2s\). The same analysis shows that as \(s\to1^-\) one recovers the classical identity \(\Delta d=-(n-1)H\), and in the fractional Allen–Cahn evolution the moving interface satisfies
\[
V_n(x)=H_s(x),
\]
yielding nonlocal curvature-driven motion for \(s<1\) and classical mean-curvature motion in the local limit [2406.14788].

Source: https://www.emergentmind.com/topics/fractional-mean-curvature