1,721,018 research outputs found
Blow-up of regular submanifolds in Heisenberg groups and applications
No full textWe obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group
Lipschitz continuity, Aleksandrov theorem and characterizations for H-convex functions
No full textIn the geometries of stratified groups, we show that H-convex functions locally bounded from above are locally Lipschitz continuous and that the class of v-convex functions exactly corresponds to the class of upper semicontinuous H-convex functions. As a consequence, v-convex functions are locally Lipschitz continuous in every stratified group. In the class of step 2 groups we characterize locally Lipschitz H-convex functions as measures whose distributional horizontal Hessian is positive semidefinite. In Euclidean space the same results were obtained by Dudley and Reshetnyak. We prove that a continuous H-convex function is a.e. twice differentiable whenever its second horizontal derivatives are Radon measures
Convexity in Carnot groups
No full textWe give an account of recent results and open questions related to the notion of convexity in Carnot groups
Characteristic points, rectifiability and perimeter measure on stratified groups
No full textWe establish an explicit formula between the perimeter measure of a domain with differentiable boundary and the (Q-1)-dimensional spherical Hausdorff measure restricted to its boundary, when the ambient space is a stratified group endowed with a sub-Riemannian structure. The spherical Hausdorff measure is built with respect to an arbitrary homogeneous distance and the integer Q denotes the Hausdorff dimension of the group with respect to its Carnot-Carathéodory distance. Our formula implies that the perimeter measure of a bounded domain with differentiable boundary is less than or equal to the (Q−1)-dimensional spherical Hausdorff measure of its boundary up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo and Nhieu. The same formula for the perimeter measure also provides an explicit expression for the optimal constants in the reciprocal estimates between perimeter measure and spherical Hausdorff measure. This result relies on two main theorems. The first one is a ``negligibility theorem" for singular points of the boundary, namely, the so-called characteristic points. We generalize this notion to submanifolds of arbitrary codimension k and we prove that the set of characteristic points is negligible with respect to the (Q-k)-dimensional spherical Hausdorff measure. The second one, is a ``blow-up theorem" for the perimeter measure of domains with differentiable boundary. We also provide an intrinsic notion of rectifiability for subsets of higher codimension, namely (G,Rk)-rectifiability. As a byproduct of the negligibility theorem, we show that rectifiable sets of codimension k with respect to the usual notion of rectifiability are (G,Rk)-rectifiable
Unrectifiability and rigidity in stratified groups
No full textIn the class of stratified groups endowed with a left invariant Carnot-Carathéodory distance, we give an algebraic characterization of purely unrectifiable groups and we study rigidity properties. The main feature of our approach is the use of a suitable area formula with respect to the Carnot-Carathéodory distance
Towards Differential Calculus in stratified groups
Full text linkIn this paper, we establish the basic tools to develop the "Calculus" associated with group-valued continuously Pansu differentiable mappings. We can ideally divide our work into five main parts. The first one develops the technical machinery on which all of our results rely. In particular, the linearization of addends appearing in the Baker-Campbell-Hausdorff formula is one of the major tools in several proofs. The second part is the characterization of graded group-valued continuously Pansu differentiable mappings through a system of nonlinear first order PDEs, namely, the "contact equations". This first result requires a preliminary characterization of absolutely continuous curves in graded groups where the contact equations play a central role. Through this approach we also establish a quantitative estimate for the Pansu difference quotient, that is crucial to obtain the mean value inequality. It is important to emphasize the potential of contact equations to tackle Lipschitz extension problems and embedding problems of surfaces modeled on a fixed group. The third part corresponds to the mean value inequality and its consequences. The first of these ones is the inverse mapping theorem, that is used in turn to achieve the rank theorem for graded group-valued continuously Pansu differentiable mappings. Another important application of the mean value inequality is in the proof of the implicit function theorem for graded group-valued continuously Pansu differentiable mappings. This is the central part of this work, since it represents our initial motivation. On the other hand, both the implicit function theorem and the rank theorem also require some algebraic conditions on the Pansu differential. These ones lead us to the fourth part of the paper, where we study factorizations of stratified groups into inner semidirect products. In fact, not all homogeneous group homomorphisms induce a natural splitting of the group. On the other hand, this splitting is a necessary condition to represent sets defined by regular mappings as intrinsic graphs. This is connected to the fifth part of the paper, where we introduce intrinsically regular sets modeled on groups, as a natural consequence of both the rank theorem and the implicit function theorem. Notice that in the case of commutative stratified groups, we obtain the classical notion of manifold. We show that these sets admit an intrinsic tangent cone at every point. On the other hand, the metric structure of regular images may differ from the one of regular level sets, since factorization induced by the intrinsic tangent cone is not "commutative". Finally, we provide some examples, where our tools can be used. We show that all Legendrian submanifolds of graded groups can be characterized as regular images modeled on Euclidean spaces. An example of new geometry where our results can be applied is the real 6-dimensional complexified Heisenberg group. Here we classify all possible intrinsically regular sets, seen either as images or as level sets
On a general coarea inequality and applications
No full textWe prove a coarea inequality for Lipschitz maps between stratified groups. As a consequence we obtain a Sard-type theorem and the nonexistence of nontrivial coarea formulae between Heisenberg groups.
In the case of real valued Lipschitz maps on the Heisenberg group we get a coarea formula using the Q-1 spherical Hausdorff measure restricted to level sets, where Q is the homogeneous dimension of the Group
Spherical Hausdorff measure of submanifolds in Heisenberg groups
No full textWe present the explicit formula relating the spherical Hausdorff measure and the Riemannian surface measure of submanifolds in the Heiseneberg group, then discussing some consequences. Complete proofs and additional results are given in a subsequent paper
Contact equations, Lipschitz extensions and isoperimetric inequalities
No full textWe characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Rumin complex. We introduce a class of 2-step groups, satisfying some abstract geometric conditions and we show that Lipschitz mappings taking values in these groups and defined on subsets of the plane admit Lipschitz extensions. We present several examples of these groups, called Allcock groups, observing that their horizontal distribution may have any codimesion. Finally, we show how these Lipschitz extensions theorems lead us to quadratic isoperimetric inequalities in all Allcock groups
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