[Paper Review] Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups

Francescopaolo Montefalcone|ArXiv.org|Oct 29, 2009

Geometric Analysis and Curvature Flows51 references3 citations

TL;DR

This paper establishes isoperimetric, Sobolev, and Poincaré inequalities on smooth hypersurfaces within sub-Riemannian Carnot groups by developing a horizontal Coarea Formula, a blow-up theorem valid at characteristic points, and a generalized 1st variation formula for the horizontal perimeter. The key contribution is a sharp isoperimetric inequality involving the horizontal mean curvature, which implies related Sobolev and Poincaré-type inequalities on compact hypersurfaces with controlled characteristic sets.

ABSTRACT

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [63] and, independently by Allard [1]. In particular, from this result one may deduce some related Sobolev-type inequalities; see Section 7. The strategy of the proof is inspired by the classical one. In particular, we shall begin by proving some linear isoperimetric inequalities. Once this is proven, one can deduce a local monotonicity formula and then conclude the proof by a covering argument. We stress however that there are many differences, due to our different geometric setting. Some of the tools which have been developed ad hoc in this paper are, in order, a ``blow-up'' theorem, which also holds for characteristic points, and a smooth Coarea Formula for the HS-gradient; see Section 3 and Section 4. Other tools are the horizontal integration by parts formula and the 1st variation of the H-perimeter already developed in [68], [69], and here generalized to hypersurfaces having non-empty characteristic set. Some natural applications of these results are in the study of minimal and constant horizontal mean curvature hypersurfaces. Moreover we shall prove some purely horizontal, local and global Poincaré-type inequalities as well as some related facts and consequences; see Section 4 and Section 5.

Motivation & Objective

• To extend classical isoperimetric, Sobolev, and Poincaré inequalities to the sub-Riemannian setting of Carnot groups.
• To establish these inequalities for C^2-smooth compact hypersurfaces with or without boundary, including those with non-empty characteristic sets.
• To develop geometric tools such as a horizontal Coarea Formula for the H-gradient and a blow-up theorem valid at characteristic points.
• To generalize the 1st variation formula for the H-perimeter to hypersurfaces with non-vanishing characteristic sets.
• To derive global and local Poincaré-type inequalities and related Sobolev embeddings on hypersurfaces in Carnot groups.

Proposed method

• Adapt the classical proof strategy of Michael and Simon and Allard by proving linear isoperimetric inequalities first.
• Introduce a horizontal Coarea Formula for the H-gradient, enabling integration over level sets of horizontal distance functions.
• Develop a blow-up theorem for the horizontal perimeter that holds even at characteristic points, crucial for local analysis.
• Use a weak monotonicity formula derived from the linear isoperimetric inequality and a covering argument to prove the main isoperimetric inequality.
• Apply the strong linear inequality and its monotonicity formula to derive asymptotic behavior of the H-perimeter and global inequalities.
• Generalize the 1st variation of the H-perimeter to hypersurfaces with non-empty characteristic sets, enabling analysis at singular points.

Experimental results

Research questions

• RQ1 Can a sharp isoperimetric inequality be established for C^2-smooth hypersurfaces in k-step Carnot groups, incorporating the horizontal mean curvature?
• RQ2 How can the classical proof strategy of Michael and Simon be adapted to the sub-Riemannian setting with non-integrable distributions?
• RQ3 What is the role of characteristic points in the isoperimetric and Sobolev theory on sub-Riemannian hypersurfaces?
• RQ4 Can global and local Poincaré-type inequalities be derived from the isoperimetric inequality in this geometric setting?
• RQ5 What are the precise Sobolev embeddings on hypersurfaces in Carnot groups, and how do they depend on the homogeneous dimension Q?

Key findings

• An isoperimetric inequality is proven for C^2-smooth compact hypersurfaces S in a k-step Carnot group, involving the horizontal mean curvature and the homogeneous (Q-1)-dimensional Hausdorff measure.
• A global Sobolev-type inequality is established: for every ψ ∈ C^1_0(S), the L^{(Q-1)/(Q-2)}-norm of ψ is bounded by a constant times the sum of the L^1-norm of |ψ| and the L^1-norm of |grad_H S ψ|.
• For p ∈ [1, Q-1[, a Sobolev embedding holds: the L^q(S)-norm of ψ is bounded by a constant times the sum of the L^p(S)-norm and the L^p(S)-norm of |grad_H S ψ| for all q ∈ [p, p*].
• When p = Q-1, the L^q(S)-norm of ψ is bounded for all q ∈ [Q-1, ∞[, with the same control on the norms.
• A strong linear isoperimetric inequality is derived, which implies a monotonicity formula for the H-perimeter, and this formula is used to analyze the asymptotic behavior of the H-perimeter near points.
• The paper proves that under assumptions (H) or (H2), the H-perimeter of the boundary of a relatively compact open set U ⊂ S is controlled by a constant multiple of the total variation of the H-gradient of the distance function.