We present different types of rotational symmetries for distances in homogeneous groups, showing that the area formula for the associated spherical measure takes a simple form.
A study of measure-theoretic area formulas Leccese, Giacomo Maria; Magnani, Valentino
Annali di matematica pura ed applicata,
06/2022, Volume:
201, Issue:
3
Journal Article
Peer reviewed
Open access
We present a complete study of measure-theoretic area formulas in metric spaces, providing different measurability conditions.
We prove an integral formula for the spherical measure of hypersurfaces in equiregular sub-Riemannian manifolds. Among various technical tools, we establish a general criterion for the uniform ...convergence of parametrized sub-Riemannian distances, and local uniform asymptotics for the diameter of small metric balls.
Let
f
be a Lipschitz map from a subset
A
of a stratified group to a Banach homogeneous group. We show that directional derivatives of
f
act as homogeneous homomorphisms at density points of
A
outside ...a
σ
-porous set. At all density points of
A
, we establish a pointwise characterization of differentiability in terms of directional derivatives. These results naturally lead us to an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon–Nikodym property.
In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian ...distance. We show how the standard notion of sectional curvature adapts to our framework, but it cannot be defined everywhere and it is unbounded on suitable sequences of planes. The vanishing of the distance precisely occurs along this sequence of planes, so that the degenerate Riemannian distance appears in connection with an unbounded sectional curvature. In the 2005 paper by Michor and Mumford, this phenomenon was first observed in some specific Fréchet manifolds.
A general approach to compute the spherical measure of submanifolds in homogeneous groups is provided. We focus our attention on the
homogeneous tangent space
, that is a suitable weighted algebraic ...expansion of the submanifold. This space plays a central role for the existence of blow-ups. Main applications are area-type formulae for new classes of
C
1
smooth submanifolds and the equality between spherical measure and Hausdorff measure on all horizontal submanifolds.
We present a new blow-up method that allows for establishing the first general formula to compute the perimeter measure with respect to the spherical Hausdorff measure in noncommutative nilpotent ...groups. This result leads us to an unexpected relationship between the area formula with respect to a distance and the profile of its corresponding unit ball.
We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these ...mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.
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