Horizontal Mean Curvature Research Articles

A characterization of gauge balls in ℍ n by horizontal curvature

Abstract In this paper, we aim at identifying the level sets of the gauge norm in the Heisenberg group ℍ n {{\mathbb{H}^{n}}} via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in ℍ 1 {\mathbb{H}^{1}} under an assumption on the location of the singular set, and in ℍ n {\mathbb{H}^{n}} for n ≥ 2 {n\geq 2} in the proper class of horizontally umbilical hypersurfaces.

A stochastic representation for the solution of approximated mean curvature flow

The evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with applications in IT and neurogeometry [see Citti et al. (SIAM J Imag Sci 9(1):212–237, 2016)]. Unfortunately this equation is difficult to study, since the horizontal normal is not always well defined. To overcome this problem the Riemannian approximation was introduced. In this article we obtain a stochastic representation of the solution of the approximated Riemannian mean curvature using the Riemannian approximation and we will prove that it is a solution in the viscosity sense of the approximated mean curvature flow, generalizing the result of Dirr et al. (Commun Pure Appl Math 9(2):307–326, 2010).

On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.

Heat content and horizontal mean curvature on the Heisenberg group

ABSTRACTWe identify the short time asymptotics of the sub-Riemannian heat content for a smoothly bounded domain in the first Heisenberg group. Our asymptotic formula generalizes prior work by van den Berg–Le Gall and van den Berg–Gilkey to the sub-Riemannian context, and identifies the first few coefficients in the sub-Riemannian heat content in terms of the horizontal perimeter and the total horizontal mean curvature of the boundary. The proof is probabilistic, and relies on a characterization of the heat content in terms of Brownian motion.

On the horizontal mean curvature flow for axisymmetric surfaces in the Heisenberg group

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.

Sub-Riemannian heat kernels and mean curvature flow of graphs

We introduce a sub-Riemannian analogue of the Bence–Merriman–Osher algorithm (Merriman et al., 1992 [42]) and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L.C. Evans (1993) [27] in the Euclidean setting and is based on new results on the relation between sub-Riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs.

A geometric inequality for stable solutions of semilinear elliptic problems in the Engel group

We prove that, if E is the Engel group and u is a stable solution of ∆Eu = f(u), then ˆ {∇Eu 6=0}  |∇Eu|2 {( p + 〈 (Hu) ν, v 〉 |∇Eu| )2 + h } − J   η ≤ ˆ E |∇Eη||∇Eu| for any test function η ∈ C∞ 0 (E). Here above, h is the horizontal mean curvature, p is the imaginary curvature and J := 2(X3X2uX1u−X3X1uX2u) + (X4u)(X1u−X2u) This can be interpreted as a geometric Poincare inequality, extending the work of [21, 22, 13] to stratified groups of step 3. As an application, we provide a non-existence result.

Local Monotonicity and Isoperimetric Inequality on Hypersurfaces in Carnot groups

Let G be a k-step Carnot group of homogeneous dimension Q. Later on we shall present some of the results recently obtained in [32] and, in particular, an intrinsic isoperimetric inequality for a C2-smooth compact hypersurface S with boundary @S. We stress that S and @S are endowed with the homogeneous measures n????1 H and n????2 H , respectively, which are actually equivalent to the intrinsic (Q - 1)-dimensional and (Q - 2)-dimensional Hausdor measures with respect to a given homogeneous metric % on G. This result generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [29] and Allard [1], independently. One may also deduce some related Sobolev-type inequalities. The strategy of the proof is inspired by the classical one and will be discussed at the rst section. After reminding some preliminary notions about Carnot groups, we shall begin by proving a linear isoperimetric inequality. The second step is a local monotonicity formula. Then we may achieve the proof by a covering argument. We stress however that there are many dierences, due to our non-Euclidean setting. Some of the tools developed ad hoc are, in order, a \blow-up theorem, which holds true also for characteristic points, and a smooth Coarea Formula for the HS-gradient. Other tools are the horizontal integration by parts formula and the 1st variation formula for the H-perimeter n????1 H already developed in [30, 31] and then generalized to hypersurfaces having non-empty characteristic set in [32]. These results can be useful in the study of minimal and constant horizontal mean curvature hypersurfaces in Carnot groups.

Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach

We study evolution by horizontal mean curvature flow in sub-Riemannian geometries by using stochastic approach to prove the existence ofa generalized evolution in these spaces. In particular we show that the valuefunction of suitable family of stochastic control problems solves in the viscositysense the level set equation for the evolution by horizontal mean curvature flow.

Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1

In the first Heisenberg group H 1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C 2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C 2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H 1 the only stable C 2 Hminimal entire graphs, with empty characteristic locus, are the vertical planes.

Constant mean curvature surfaces in sub-Riemannian geometry

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the Euler-Lagrange equations for critical points for the associated variational problems. Using the Euler-Lagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II0, for vertically rigid spaces and, as a first application, use II0 to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in Carnot groups cannot be locally strictly horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature.

Geometric second derivative estimates in Carnot groups and convexity

We prove some new a priori estimates for H 2-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group $${\mathbb{G}}$$ . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature $${\mathcal{H}}$$ of ∂Ω. As a consequence of our bounds we show that if $${\mathbb{G}}$$ has step two, then for any smooth H 2-convex function in $$\Omega \subset {\mathbb{G}}$$ vanishing on ∂Ω one has $$\sum \limits _{i,j=1} ^m \int \limits_\Omega ([X_i,X_j]u)^2 \, dg \, \leq \, \frac{4}{3} \int \limits_{\partial \Omega} \mathcal H\ |\nabla_H u|^2\, d\sigma_H$$ .

Horizontal Connection and Horizontal Mean Curvature in Carnot Groups

In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli–Garofalo–Nhieu and Pauls who recently introduced sub–Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction (projection) of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and (intrinsic) regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.

On some sub-Riemannian objects in hypersurfaces of sub-Riemannian manifolds

We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then there exists at least a piecewise smooth horizontal curve in this hypersurface connecting any two given points in it. In any sub-Riemannian manifold, we obtain the sub-Riemannian version of the fundamental theorem of Riemannian geometry: there exists a unique nonholonomic connection which is completely determined by the sub-Riemannian structure and is “symmetric” and compatible with the sub-Riemannian metric. We use this nonholonomic connection to study horizontal mean curvature of hypersurfaces.

Real hypersurfaces in Kähler manifolds

Building on work by S. M. Webster \ref[J. Differential Geom. 13 (1978), no. 1, 25--41; MR0520599 (80e:32015)] the author studies the geometry of the second fundamental form of a real hypersurface in a Kahler manifold. As an application he proves that a compact strictly pseudoconvex hypersurface $M\subsetC^n$ is isometric to a sphere provided that $M$ has constant horizontal mean curvature and the CR structure $T_{1,0}(M)$ is parallel in $T^{1,0}(C^n)$.