Unexpected phenomena for mean curvature functionals in the Heisenberg group

Abstract: The Euclidean paradigm that spheres optimize mean curvature variational problems breaks down in the sub-Riemannian Heisenberg group: neither the Pansu sphere nor the Korányi sphere is optimal for the variational problems associated with the Minkowski and Heintze-Karcher inequalities. Motivated by this phenomenon, we develop a variational theory for geometric problems driven by the horizontal mean curvature, focusing on the total mean curvature functional and the related Minkowski inequality. To investigate this phenomenon, we establish first and second variation formulas for general mean curvature functionals in arbitrary Riemannian manifolds, and then obtain corresponding formulas in Heisenberg groups through a Riemannian approximation scheme. We subsequently specialize to the optimization of total mean curvature under area constraint in the first Heisenberg group, introducing suitable notions of non-characteristic stationarity and stability. We identify a new one-parameter family of rotationally invariant critical surfaces, which we call Pansu-Minkowski spheres. Among them, we show that a distinguished member, the optimal Pansu-Minkowski sphere, emerges as the unique critical point of the Minkowski quotient, and uniquely minimizes it among Pansu-Minkowski spheres. We prove non-characteristic stability and local minimality of Pansu-Minkowski spheres under rotationally invariant perturbations, while showing their instability under unrestricted perturbations.

• The paper develops first- and second-variation formulas for functionals of horizontal mean curvature using Riemannian approximation, including a general sub-Riemannian Jacobi operator framework.
• The authors show that Pansu and Korányi spheres are neither critical nor optimal for the Minkowski and Heintze–Karcher quotients, with explicitly nonzero scaling derivatives demonstrating this failure.
• The paper classifies rotationally invariant Pansu–Minkowski spheres, identifies S₁⁄₄ as the unique critical member for the Minkowski quotient with value (18π)¹⁄³, and proves rotational stability but instability under unrestricted perturbations.

Overview

This paper by Fogagnolo, Pinamonti, and Verzellesi develops a variational theory for geometric functionals driven by the horizontal mean curvature $H^\nu$ in the Heisenberg group $\mathbb H^n$, and applies it to the total mean curvature functional in $\mathbb H^1$. The central finding is a breakdown of the Euclidean paradigm: neither the Pansu sphere nor the Korányi sphere is optimal, or even critical, for the variational problems associated with the Minkowski and Heintze–Karcher inequalities. The authors identify a new one-parameter family of rotationally invariant critical surfaces — the Pansu–Minkowski spheres — of which exactly one member is critical for the Minkowski quotient, and they establish stability under rotationally invariant perturbations but instability under unrestricted perturbations.

Non-optimality of canonical spheres

The functionals of interest are the Minkowski quotient

$$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$

where $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$ is the total horizontal mean curvature, and the Heintze–Karcher quotient $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$ involving the total inverse mean curvature. In Euclidean space, spheres uniquely optimize both; in $\mathbb H^n$ the natural candidates are the Pansu sphere (the unique closed constant-horizontal-mean-curvature surface up to dilations) and the Korányi sphere.

The first main result shows both candidates fail. Along the family of rescaled Pansu spheres $S^R$, the authors compute explicitly that at $R=1$,

$$Q_{\mathrm{mink}}^\nu(S^1)=2\pi^{2/3},\qquad \left.\frac{d}{dR}\right|_{R=1}Q_{\mathrm{mink}}^\nu(S^R)=\pi^{2/3}\neq 0,$$

and

$\mathbb H^n$0

For the Korányi sphere, the corresponding derivatives are $\mathbb H^n$1 and $\mathbb H^n$2, again nonzero. The mechanism is structural rather than accidental: the quotients are invariant under intrinsic dilations $\mathbb H^n$3 but not under Euclidean dilations, so suitably rescaled deformations strictly decrease them. Moreover, by composing these deformations with intrinsic rescalings that fix area or enclosed volume, the authors show the two spheres are not even stationary for the constrained problems. Two consequences follow immediately: optimal configurations for the main sub-Riemannian curvature-driven problems lack the symmetry of their Euclidean counterparts, and a hypothetical Heintze–Karcher inequality cannot serve as a direct route to an Aleksandrov-type rigidity theorem in this setting.

Variation formulas

The technical core of the paper consists of first and second variation formulas for general functionals of the form

$\mathbb H^n$4

along non-characteristic variations, i.e., variations supported away from the characteristic set $\mathbb H^n$5. These are established in full generality for arbitrary closed hypersurfaces in $\mathbb H^n$6 and arbitrary smooth $\mathbb H^n$7, via a Riemannian approximation scheme on $\mathbb H^n$8 where the vertical direction is penalized by a factor $\mathbb H^n$9. As a prerequisite, the authors derive corresponding Riemannian formulas for arbitrary variations in arbitrary ambient manifolds, which appear to be new at this level of generality since prior literature treated only special functionals, restricted variation classes, or additional curvature assumptions.

The approximation argument requires careful tracking of the $\mathbb H^1$0 behavior of the extrinsic geometry: the second fundamental form, its cubic trace, the Ricci curvature of the approximating metrics, and the relation between the Riemannian Laplacian and the modified horizontal tangential Laplacian $\mathbb H^1$1, where $\mathbb H^1$2 is the fundamental function. The resulting sub-Riemannian formula expresses the second variation through a self-adjoint operator $\mathbb H^1$3 built from the horizontal Jacobi operator $\mathbb H^1$4 and the horizontal shape operator $\mathbb H^1$5.

Specializing to $\mathbb H^1$6 in $\mathbb H^1$7, the first variation simplifies to

$\mathbb H^1$8

with $\mathbb H^1$9 the normal velocity component. Stationarity under area-preserving non-characteristic variations is then equivalent to the Euler–Lagrange equation

$$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$0

and to stationarity of the penalized functional $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$1. Stability transfers between the constrained problem and the penalized functional along first-order area-preserving variations, via a standard Lagrange multiplier reduction adapted to the non-characteristic setting.

Pansu–Minkowski spheres

Solving the Euler–Lagrange equation within the class of rotationally invariant surfaces reduces the problem to a scalar ODE for the profile curve. The analysis proceeds in three steps: degenerate cases ($$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$2) yield only planes, cylinders, and paraboloid-type profiles; a priori bounds force $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$3 and rule out toroidal topology for mean convex solutions; and a clever substitution converts the ODE into a quadratic equation whose solvability under verticality boundary conditions forces $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$4. The outcome is a complete classification: every rotationally invariant, closed, mean convex critical point is, up to dilations and vertical translations, the surface of revolution of the profile

$$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$5

for $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$6. These are the Pansu–Minkowski spheres $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$7; the endpoint $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$8 recovers the Pansu sphere.

Two features distinguish this classification from the Euclidean case, where the sphere is the unique equilibrium configuration. First, there are infinitely many critical points. Second, although the Pansu sphere fails to be stationary for $$Q_{\mathrm{mink}}^\nu(S) = \left(\mathcal M^\nu(S)\right)^{-2/3}\sigma^\nu(S),$$9 under unrestricted area-preserving variations, it is stationary along non-characteristic ones — characteristic points therefore genuinely alter the variational landscape, in contrast with the isoperimetric problem where the Pansu sphere remains the unique closed constant-mean-curvature competitor even among surfaces with isolated characteristic points.

Within the family, the constrained and unconstrained problems decouple at first order: every critical point of $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$0 is an area-preserving critical point of $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$1, but not conversely. An explicit computation of the quotient along the family, reparametrized via $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$2, shows that

$\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$3

with equality if and only if $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$4. This distinguished member, the optimal Pansu–Minkowski sphere $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$5, is the unique critical point of the Minkowski quotient among all rotationally invariant surfaces, characterized also by the rigidity condition $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$6.

Stability and instability

Evaluating the second variation of the penalized functional at $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$7, the zero-order coefficient simplifies to $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$8, which attains its minimum $\mathcal M^\nu(S)=\int_S H^\nu\,d\sigma^\nu$9 at $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$0. For rotationally invariant perturbations ($$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$1 independent of $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$2), a sharp pointwise bound on the first-order term yields the coercive estimate

$$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$3

Notably, this bound requires no first-order area constraint, making it stronger than ordinary stability — Euclidean spheres do not enjoy such a property for the analogous penalized isoperimetric functional. Combined with a Taylor expansion argument controlling higher-order terms, this gives local minimality of every $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$4 among sufficiently small rotationally invariant horizontally normal graphs preserving area.

The picture reverses completely without the symmetry restriction. Choosing angular test functions $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$5, which are automatically first-order area-preserving because the angular average vanishes, the second variation acquires a negative contribution proportional to $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$6:

$$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$7

which dominates for $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$8 large. Hence every Pansu–Minkowski sphere is unstable for $$Q_{\mathrm{hk}}^\nu(S)=|\Omega(S)|^{-1}\mathcal A^\nu(S)$$9 along area-preserving non-characteristic variations. Consequently, any global minimizer of the Minkowski quotient, should one exist, cannot be rotationally invariant. This mirrors phenomena observed in the sub-Riemannian isodiametric problem and casts doubt on both existence and uniqueness of minimizers.

Limitations and open questions

Several restrictions qualify the results. All variational statements concern non-characteristic variations; the behavior at characteristic points, where the horizontal geometry collapses, is addressed only indirectly, and the failure of the Pansu sphere to be stationary under unrestricted variations demonstrates that this restriction is essential rather than technical. The classification of critical points and the stability analysis are confined to rotationally invariant surfaces in $\mathbb H^n$0; nothing is proved about non-symmetric critical points or higher-dimensional Heisenberg groups. The Heintze–Karcher problem is treated only at the level of negative results, though the authors observe that the Pansu sphere does satisfy the corresponding Euler–Lagrange equation under non-characteristic volume-preserving variations. The global minimization of $\mathbb H^n$1 remains open: it is natural to conjecture that $\mathbb H^n$2 minimizes the quotient among rotationally invariant competitors, but the instability result leaves open whether any minimizer exists outside the symmetric class, and whether the sharp constant $\mathbb H^n$3 is realized globally. More broadly, the paper leaves open how to formulate sharp mean-curvature-driven inequalities once symmetry ceases to be a reliable guide.

Conclusion

The paper establishes that the Euclidean identification of optimal shapes across isoperimetric, Minkowski, and Heintze–Karcher type problems fails in the Heisenberg group: the Pansu and Korányi spheres are not even critical points of the relevant quotients. Through general first and second variation formulas obtained by Riemannian approximation, the authors classify all rotationally invariant critical points of the total mean curvature under area constraint, isolate the optimal Pansu–Minkowski sphere $\mathbb H^n$4 as the unique critical point of the Minkowski quotient, and prove a sharp dichotomy — local minimality under rotationally invariant perturbations, instability under unrestricted ones. These results indicate that characteristic points and symmetry constraints play structurally decisive roles in sub-Riemannian variational geometry, and they reframe the search for sharp inequalities as a problem in which symmetric configurations need not be extremal.