Generalised Mean Curvature in Extended Geometries

• Generalised mean curvature is an extension of classical curvature, defined relative to ambient structures such as exact Courant algebroids, weighted functionals, and nonsmooth variational formulations.
• It is constructed via diverse frameworks, including generalized second fundamental forms, weighted density corrections, and approximation schemes for variational problems in geometry.
• Its applications span modern geometric analysis from hypersurface theory and curvature flows to complex-valued invariants in Riemann–Cartan and generalized geometric settings.

Generalised mean curvature is a family of extensions of classical mean curvature beyond the standard hypersurface setting in a Riemannian manifold. In the most direct contemporary usage, it is introduced in the transfer of tools from semi-Riemannian immersion theory to generalised geometry: for hypersurfaces inside exact Courant algebroids, it appears alongside a generalised second fundamental form, generalised exterior curvature, and generalised Gauß–Codazzi equations (Cortés et al., 16 Jul 2025). More broadly, the same expression is used for weighted curvature functionals, weak varifold mean-curvature vectors, kernel-smoothed approximations on point clouds, convex-geometric total mean curvature for nonsmooth strictly convex bodies, and torsion-corrected complex-valued surface invariants. The term is therefore not univocal; its precise meaning is determined by the ambient geometric structure, the variational problem, and the regularity class under consideration.

1. Classical reference point

The classical reference point is the mean curvature $H$ obtained as the trace of the Weingarten map, $H=\operatorname{tr}(W)$, for an oriented hypersurface with unit normal (Lee, 18 Feb 2025). In $\mathbb{R}^3$, for a $C^2$-smooth closed surface $E$, this gives $H=\frac{k_1+k_2}{2}$, and the total mean curvature is $\int_E H\,dE$ (Charytanowicz et al., 2019). In flow problems, the same quantity appears as the geometric velocity law $\frac{\partial x}{\partial t}=-Hn$ or, with alternative sign conventions, as $H=\operatorname{div}_{\Sigma_t}\nu$ for graphical hypersurfaces in Lorentzian settings (Li et al., 2018, Lira et al., 2022).

The classical object already has several structurally distinct roles. It is the first-variation term for area, the scalar controlling constant-mean-curvature rigidity, the curvature quantity entering halfspace theorems, and the trace component of the second fundamental form in immersion theory (Mazet, 2010, Edelen et al., 2013). Generalisations preserve different subsets of these roles. In some frameworks the extension remains a trace-type quantity; in others it is the correct first-variation term for a weighted functional, an integral invariant defined without pointwise curvature, or a weak vector field recovered from first variation.

2. Generalised mean curvature in exterior generalised geometry

In "Exterior Generalised Geometry" (Cortés et al., 16 Jul 2025), the setting is an exact Courant algebroid $E\to M$ together with an immersion $H=\operatorname{tr}(W)$0. The paper states that there is a well-known construction of an exact Courant algebroid $H=\operatorname{tr}(W)$1, the pullback of $H=\operatorname{tr}(W)$2, and explains the pullback of generalised metrics and divergence operators. Assuming that $H=\operatorname{tr}(W)$3 is a hypersurface, it develops the notion of generalised exterior curvature, introducing the generalised second fundamental form and the generalised mean curvature (Cortés et al., 16 Jul 2025).

The same abstract places generalised mean curvature inside a larger hypersurface calculus. Generalised versions of the Gauß–Codazzi equations are obtained, and the formalism is applied to the constraint equations for the initial value formulation of the generalised Einstein equations. Further applications include a generalised geometry version of the fundamental theorem for hypersurfaces, the restriction of generalised Kähler and hyper-Kähler structures to submanifolds compatible with the generalised almost complex structure, and the characterisation of exact semi-Riemannian Courant algebroids which are flat with respect to the canonical generalised connection (Cortés et al., 16 Jul 2025). A related paper constructs that canonical generalised Levi-Civita connection from a generalised metric $H=\operatorname{tr}(W)$4 and a divergence operator $H=\operatorname{tr}(W)$5 on an exact Courant algebroid, together with decompositions of the associated generalised curvature tensors (Cortés et al., 23 Jul 2025).

At the level of the published abstract, the concept is therefore specified structurally rather than by an explicit displayed formula. The announced hierarchy is: pullback Courant algebroid, pullback generalised metric and divergence, generalised second fundamental form, generalised mean curvature, and then generalised Gauß–Codazzi theory (Cortés et al., 16 Jul 2025). This suggests that, in exterior generalised geometry, generalised mean curvature is intended as the trace-level hypersurface invariant compatible with the pair $H=\operatorname{tr}(W)$6 and with the canonical connection formalism.

3. Weighted, prescribed, and conformal extensions

A major classical extension replaces ordinary mean curvature by a weighted quantity. In "Constant mean curvature, flux conservation, and symmetry" (Edelen et al., 2013), the density $H=\operatorname{tr}(W)$7 leads to the weighted mean curvature

$H=\operatorname{tr}(W)$8

where $H=\operatorname{tr}(W)$9 is the usual mean curvature and $\mathbb{R}^3$0 is the unit normal. Hypersurfaces with constant weighted mean curvature $\mathbb{R}^3$1 are critical points of the functional $\mathbb{R}^3$2, and this weighted curvature is the quantity governing the generalised flux conservation law for caps and spines in the presence of Killing fields (Edelen et al., 2013). In this usage, generalisation means that the first-variation formula is modified by a density rather than by a change of codomain.

A closely related conformal reinterpretation appears in the translating mean curvature equation. For a graph $\mathbb{R}^3$3 over a domain $\mathbb{R}^3$4, the equation

$\mathbb{R}^3$5

is exactly the condition that the graph of $\mathbb{R}^3$6 be minimal in the conformal product manifold $\mathbb{R}^3$7 with metric $\mathbb{R}^3$8 (Zhou, 2019). The same paper defines a conformal area functional $\mathbb{R}^3$9, proves that its generalized solutions are characterised by local perimeter minimality of the subgraph in $C^2$0, and thereby turns the Dirichlet problem into a generalized mean curvature/perimeter problem in a conformal product manifold (Zhou, 2019).

Another extension prescribes mean curvature as a function of the Gauss map. If $C^2$1, an oriented immersed hypersurface $C^2$2 is an $C^2$3-hypersurface when

$C^2$4

with $C^2$5 the Gauss map (Bueno et al., 2018). This theory contains constant mean curvature as the special case of constant $C^2$6, and it also contains self-translating solitons, which are described as hypersurfaces of constant weighted mean curvature for the density $C^2$7 with $C^2$8 (Bueno et al., 2018). Here the generalisation is directional: curvature is prescribed by normal direction rather than fixed as a constant.

4. Weak, approximate, and nonsmooth formulations

In geometric measure theory, generalised mean curvature is fundamentally a first-variation object. For a rectifiable $C^2$9-varifold $E$0 with bounded first variation, there exist a generalized mean curvature vector $E$1, a singular boundary variation measure $E$2, and a unit vector field $E$3 such that

$E$4

for every compactly supported $E$5 vector field $E$6 (White, 2019). White’s theory of integral Brakke flow with boundary takes this as the basic weak formulation; the boundary is then re-expressed through a vector field $E$7 on $E$8, and the Brakke inequality is imposed on time slices lying in the corresponding class $E$9 (White, 2019). In this setting, generalised mean curvature is a vector field in the interior, while the boundary contribution is encoded separately.

A different weak extension is kernel-smoothed and approximation-theoretic. In "Approximations of the mean curvature, and the Buet-Rumpf approximate mean curvature flow" (Sagueni, 8 Sep 2025), the approximate mean curvature of a $H=\frac{k_1+k_2}{2}$0-varifold $H=\frac{k_1+k_2}{2}$1 is defined by a Buet–Rumpf type formula with an inserted linear operator $H=\frac{k_1+k_2}{2}$2,

$H=\frac{k_1+k_2}{2}$3

when $H=\frac{k_1+k_2}{2}$4 (Sagueni, 8 Sep 2025). The paper identifies operator families for which this converges to the true mean curvature on smooth submanifolds and on sufficiently regular unit-density integral varifolds, extends the construction to the approximate second fundamental form, and proves comparison principles for motion of point clouds by mean curvature in both continuous and discrete cases (Sagueni, 8 Sep 2025).

Nonsmooth convex geometry produces yet another non-equivalent extension. "A generalization of the total mean curvature" (Charytanowicz et al., 2019) derives a special formula for the total mean curvature of an ovaloid and uses it to extend the notion to boundaries of strictly convex sets in $H=\frac{k_1+k_2}{2}$5. The extension is expressed by an integral over the region bounded by the pedal surface, and the support-function identity

$H=\frac{k_1+k_2}{2}$6

is proved for a strictly convex set $H=\frac{k_1+k_2}{2}$7 with support function $H=\frac{k_1+k_2}{2}$8 (Charytanowicz et al., 2019). In this formulation, the generalised quantity is not a local trace of a second fundamental form; it is an integral invariant that still agrees with $H=\frac{k_1+k_2}{2}$9 in the smooth ovaloid case (Charytanowicz et al., 2019).

5. Torsion, antisymmetry, and complex-valued curvature

In Riemann–Cartan geometry, the ambient connection is metric-compatible but has torsion, and the second fundamental form is no longer symmetric. For a hypersurface with unit normal $\int_E H\,dE$0, the torsion 2-form is

$\int_E H\,dE$1

and the antisymmetric part of the second fundamental form satisfies

$\int_E H\,dE$2

(Lee, 18 Feb 2025). The paper "Complex-valued extension of mean curvature for surfaces in Riemann-Cartan geometry" packages this extra datum into the complex-valued scalar

$\int_E H\,dE$3

where $\int_E H\,dE$4 is the usual mean curvature and $\int_E H\,dE$5 is the Hodge dual of the torsion 2-form on the surface (Lee, 18 Feb 2025).

This construction changes the ontology of mean curvature itself. The real part still measures normal bending, while the imaginary part measures torsional twisting induced by the ambient connection (Lee, 18 Feb 2025). In Weitzenböck geometry, if the Gauss map with respect to a global orthonormal frame is $\int_E H\,dE$6, then

$\int_E H\,dE$7

so the complex-valued generalised mean curvature becomes a divergence–curl package for the Gauss map (Lee, 18 Feb 2025). The paper further shows that $\int_E H\,dE$8 interacts with the Hopf differential, the third fundamental form, holomorphicity criteria, and conformality of the Gauss map. In this usage, generalisation is driven by torsion and by the failure of symmetry of the second fundamental form.

6. Generalised flows, higher-curvature hierarchies, and scope of the term

In geometric evolution, "generalized mean curvature flow" often means that the normal speed is an arbitrary strictly monotone function of $\int_E H\,dE$9. For a closed two-dimensional surface in $\frac{\partial x}{\partial t}=-Hn$0, the evolution law

$\frac{\partial x}{\partial t}=-Hn$1

with $\frac{\partial x}{\partial t}=-Hn$2 a smooth strictly increasing function, simultaneously includes mean curvature flow, inverse mean curvature flow, powers of mean curvature flow, and powers of inverse mean curvature flow (Binz et al., 2020). The same paper derives the coupled parabolic system

$\frac{\partial x}{\partial t}=-Hn$3

and proves optimal-order $\frac{\partial x}{\partial t}=-Hn$4-error bounds for an ESFEM plus linearly implicit BDF discretisation (Binz et al., 2020). Here the adjective “generalized” refers to the nonlinear dependence of the speed on mean curvature.

A related, but distinct, enlargement of the curvature vocabulary appears in the hierarchy of higher mean curvatures $\frac{\partial x}{\partial t}=-Hn$5. "New Hsiung-Minkowski identities" (Albuquerque, 2021) establishes integral identities for a closed hypersurface in an orientable Riemannian manifold endowed with a vector field $\frac{\partial x}{\partial t}=-Hn$6, with $\frac{\partial x}{\partial t}=-Hn$7 the normalized mean curvature and $\frac{\partial x}{\partial t}=-Hn$8 the normalized second and third mean curvatures. The specialised identities for a position vector field relate weighted integrals of $\frac{\partial x}{\partial t}=-Hn$9, $H=\operatorname{div}_{\Sigma_t}\nu$0, and the $H=\operatorname{div}_{\Sigma_t}\nu$1, together with ambient Ricci and curvature terms (Albuquerque, 2021). Although these $H=\operatorname{div}_{\Sigma_t}\nu$2 are not all called “generalised mean curvature” in that paper, they show that modern curvature theory often treats mean curvature as the first level of a larger symmetric-polynomial hierarchy rather than as an isolated scalar.

The principal misconception is therefore terminological. There is no single canonical object called generalised mean curvature across the literature. In exact Courant algebroids it belongs to hypersurface theory in generalised geometry (Cortés et al., 16 Jul 2025); in weighted geometry it is $H=\operatorname{div}_{\Sigma_t}\nu$3 (Edelen et al., 2013); in Riemann–Cartan geometry it is the complex quantity $H=\operatorname{div}_{\Sigma_t}\nu$4 (Lee, 18 Feb 2025); in varifold theory it is the first-variation vector $H=\operatorname{div}_{\Sigma_t}\nu$5 or a kernel-smoothed approximation $H=\operatorname{div}_{\Sigma_t}\nu$6 (White, 2019, Sagueni, 8 Sep 2025); in convex geometry it is an integral extension of total mean curvature to strictly convex boundaries (Charytanowicz et al., 2019); and in curvature-driven evolution it may denote a nonlinear speed law built from $H=\operatorname{div}_{\Sigma_t}\nu$7 (Binz et al., 2020). What unifies these notions is not a common formula but a common role: each is the curvature datum that remains geometrically or variationally natural after the ambient structure has been enlarged.