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Roberto Monti

Publications and source records attributed to Roberto Monti.

At least 19 recordsLinked to original sources
arXiv2025

Plateau's Problem for intrinsic graphs in the Heisenberg Group

Using a geometric construction, we solve Plateau's Problem in the Heisenberg group $\mathbb{H}^{1}$ for intrinsic graphs defined on a convex domain $D$, under a smallness condition either on the boundary $\partial D$ or on the Lipschitz boundary datum $\varphi : \partial D \to \mathbb{R}$. The proof relies on a calibration argument. We then apply these techniques to establish a new regularity result for $H$-perimeter minimizers.

math.CA↗
arXiv2025

Not all sub-Riemannian minimizing geodesics are smooth

A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a $C^2$ but not $C^3$ length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.

math.DG↗
arXiv2024

Higher order Goh conditions for singular extremals of corank 1

We prove Goh conditions of order n for strictly singular length minimizing curves of corank 1, under the assumption that the lower order intrinsic differentials of the end-point map vanish. This result relies upon the proof of an open mapping theorem for maps with non-singular nth differential.

math.DG↗
arXiv2024

Mean value formulas on surfaces in Grushin spaces

We prove (sub)mean value formulas at the point $0\inΣ$ for (sub)harmonic functions a on a hypersurface $Σ\subset\mathbb{R}^{n+1}$ where the differentiable structure and the surface measure depend on the ambient Grushin structure.

math.AP↗
arXiv2022

The isoperimetric problem for regular and crystalline norms in $\mathbb H^1$

We study the isoperimetric problem for anisotropic left-invariant perimeter measures on $\mathbb R^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $ϕ$ on the horizontal distribution. We first prove a representation formula for the $ϕ$-perimeter of regular sets and, assuming some regularity on $ϕ$ and on its dual norm $ϕ^*$, we deduce a foliation property by sub-Finsler geodesics of $\mathrm C^2$-smooth surfaces with constant $ϕ$-curvature. We then prove that the characteristic set of $\mathrm C^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize $\mathrm C^2$-smooth $ϕ$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $ϕ$, that such sub-Finsler candidate isoperimetric sets are indeed $\mathrm C^2$-smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where $ϕ$ is crystalline).

math.DG↗
arXiv2021

Nonminimality of spirals in sub-Riemannian manifolds

We show that in analytic sub-Riemannian manifolds of rank 2 satisfying a commutativity condition spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve.

math.DG↗
arXiv2019

John and uniform domains in generalized Siegel boundaries

Given the pair of vector fields $X=\partial_x+|z|^{2m}y\partial_t$ and $ Y=\partial_y-|z|^{2m}x \partial_t,$ where $(x,y,t)= (z,t)\in\mathbb{R}^3=\mathbb{C}\times\mathbb{R}$, we give a condition on a bounded domain $Ω\subset\mathbb{R}^3$ which ensures that $Ω$ is an $(ε,δ)$-domain for the Carnot-Carathéodory metric. We also analyze the Ahlfors regularity of the natural surface measure induced at the boundary by the vector fields.

math.MG↗
arXiv2019

Third order open mapping theorems and applications to the end-point map

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize the abstract theory to the study of length-minimality of sub-Riemannian strictly singular curves. We conclude with the third order analysis of a specific strictly singular extremal that is not length-minimizing.

math.DG↗
arXiv2018

A Trace theorem for Martinet--type vector fields

In $\mathbb{R}^3$ we consider the vector fields \[ X_1 =\frac{ \partial }{\partial x},\qquad X_2 =\frac{ \partial }{\partial y}+ |x|^α\frac{ \partial }{\partial z}, \] where $α\in\left[1,+\infty\right[$. Let $\mathbb{R}^3_+ =\{(x,y,z)\in\mathbb{R}^3: z\geq 0\}$ be the (closed) upper half-space and let $f\in C^1 ( \mathbb{R} ^3_+ )$ be a function such that $X_1f, X_2f \in L^ p(\mathbb{R}^3_+)$ for some $p>1$. In this paper, we prove that the restriction of $f$ to the plane $z=0$ belongs to a suitable Besov space that is defined using the Carnot-Carathéodory metric associated with $X_1$ and $X_2$ and the related perimeter measure.

math.CA↗
arXiv2017

Existence of tangent lines to Carnot-Carathéodory geodesics

We show that length minimizing curves in Carnot-Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. This is the first regularity result for length minimizers that holds with no assumption on either the space (e.g., its rank, step, or analyticity) or the curve, and it is novel even in the setting of Carnot groups.

math.OC↗
arXiv2016

Improved Lipschitz approximation of $H$-perimeter minimizing boundaries

We prove two new approximation results of $H$-perimeter minimizing boundaries by means of intrinsic Lipschitz functions in the setting of the Heisenberg group $\mathbb{H}^n$ with $n\ge2$. The first one is an improvement of a result of Monti and is the natural reformulation in $\mathbb{H}^n$ of the classical Lipschitz approximation in $\mathbb{R}^n$. The second one is an adaptation of the approximation via maximal function developed by De Lellis and Spadaro.

math.MG↗
arXiv2016

CMC spheres in the Heisenberg group

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group $H^1$. These spheres are conjectured to be the isoperimetric sets of $H^1$. We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

math.MG↗
arXiv2015

Quantitative isoperimetric inequalities in H^n

In the Heisenberg group H^n, we prove quantitative isoperimetric inequalities for Pansu's spheres, that are known to be isoperimetric under various assumptions. The inequalities are shown for suitably restricted classes of competing sets and the proof relies on the construction of sub-calibrations.

math.OC↗
arXiv2014

Isoperimetric problem in H-type groups and Grushin spaces

We study the isoperimetric problem in H-type groups and Grushin spaces, emphasizing a relation between them. We prove existence, symmetry and regularity properties of isoperimetric sets, under a symmetry assumption that depends on the dimension.

math.OC↗
arXiv2014

Height estimate and slicing formulas in the Heisenberg group

We prove a height-estimate (distance from the tangent hyperplane) for $Λ$-minima of the perimeter in the sub-Riemannian Heisenberg group. The estimate is in terms of a power of the excess ($L^2$-mean oscillation of the normal) and its proof is based on a new coarea formula for rectifiable sets in the Heisenberg group.

math.CA↗
arXiv2014

Corners in non-equiregular sub-Riemannian manifolds

We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results [4]. As an application of our main result we complete and simplify the analysis in [6], showing that in a 4-dimensional sub-Riemannian structure suggested by Agrachev and Gauthier all length-minimizing curves are smooth.

math.OC↗
arXiv2013

Extremal curves in nilpotent Lie groups

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin Maximum Principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.

math.DG↗