Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups. (Caractérisation des graphes intrinsèques co-horizontaux uniformément différentiables dans les groupes de Carnot.) (English. French summary) Zbl 07929014
Summary: In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are \(C^1_{\mathrm{H}}\) regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Hölder properties along the projections of left-invariant vector fields on the graph.
We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group.
As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.
We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group.
As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.
MSC:
| 53C17 | Sub-Riemannian geometry |
| 22E25 | Nilpotent and solvable Lie groups |
| 28A75 | Length, area, volume, other geometric measure theory |
| 49N60 | Regularity of solutions in optimal control |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 26A16 | Lipschitz (Hölder) classes |
Keywords:
Carnot groups; intrinsically \(C^1\) surfaces; co-horizontal surfaces; area formula; intrinsically differentiable functions; little Hölder functions; broad solutionsReferences:
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