Stepanov Differentiability Theorem for Intrinsic Graphs in Heisenberg Groups

Conceitos essenciais

Intrinsic Lipschitz functions defined on complementary subgroups of the Heisenberg group are intrinsically differentiable almost everywhere on the set of points where their pointwise intrinsic Lipschitz constant is finite.

Resumo

The paper proves a Stepanov-type differentiability theorem for intrinsic graphs in Heisenberg groups. The key insights are:

The authors consider intrinsic Lipschitz functions defined on complementary subgroups of the Heisenberg group Hn, where one subgroup is horizontal (abelian) and the other is vertical (normal).

They show that such intrinsic Lipschitz functions are intrinsically differentiable almost everywhere on the set of points where their pointwise intrinsic Lipschitz constant is finite.

The proof is divided into two main cases:

For maps from a horizontal subgroup to a normal subgroup, the authors reduce the problem to the classical Stepanov theorem for Carnot groups.
For maps from a normal subgroup to a horizontal subgroup, the authors develop a new geometric approach based on the properties of intrinsic cones and the structure of complementary subgroups in Hn.

The authors also provide an alternative proof for the case of codimension one, using an elegant approach inspired by the work of Malý.

The results are not limited to the Heisenberg group and can be extended to intrinsic graphs in general Carnot groups with a suitable splitting of the group.

Estatísticas

None.

Citações

"Roughly speaking, the Stepanov differentiability theorem generalizes the classical notion of differentiability by relaxing the conditions under which a function is differentiable almost everywhere, broadening the scope of functions to include those that may not exhibit pointwise regularity."
"Remarkably, in [23], the authors constructed intrinsic Lipschitz graphs of codimension 2 in Carnot groups which are not intrinsically differentiable almost everywhere thus discovering a deep connection between the notion of intrinsic differentiability and the geometry of the underlying Carnot group."

Principais Insights Extraídos De

Stepanov Differentiability Theorem for intrinsic graphs in Heisenberg groups

by Marco Di Mar... às arxiv.org 10-03-2024

https://arxiv.org/pdf/2410.01526.pdf

Perguntas Mais Profundas

What are the potential applications of the Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups and other Carnot groups?

The Stepanov differentiability theorem for intrinsic graphs in Heisenberg groups and other Carnot groups has several potential applications across various fields of mathematics and applied sciences. One significant application lies in the study of geometric measure theory, particularly in understanding the regularity properties of intrinsic Lipschitz functions and their graphs. This theorem provides a framework for establishing the differentiability of functions that may not be classically Lipschitz but still exhibit a form of regularity intrinsic to the structure of Carnot groups.
In the context of sub-Riemannian geometry, the theorem can be instrumental in analyzing the behavior of curves and surfaces within these groups, which is crucial for applications in control theory and robotics, where paths must be optimized under constraints defined by the underlying geometry. Furthermore, the results can be applied to problems in image processing and computer vision, where intrinsic graphs can model surfaces and shapes in a way that respects the geometric properties of the data.
Additionally, the theorem may find relevance in the study of partial differential equations (PDEs) defined on Carnot groups, where understanding the differentiability of solutions is essential for establishing regularity results. The insights gained from the Stepanov differentiability theorem can thus contribute to a deeper understanding of the solutions to these PDEs, particularly in non-Euclidean settings.

How can the techniques developed in this paper be extended to study the differentiability properties of more general classes of functions defined on Carnot groups?

The techniques developed in this paper can be extended to study the differentiability properties of more general classes of functions defined on Carnot groups by leveraging the intrinsic geometric structures and properties of these groups. One approach is to generalize the notion of intrinsic Lipschitz continuity and differentiability to encompass broader classes of functions, such as those that may exhibit non-standard growth conditions or are defined on more complex subsets of Carnot groups.
By employing the framework of intrinsic graphs and the associated geometric measures, researchers can analyze the behavior of functions that are not necessarily Lipschitz but still possess some form of controlled growth or regularity. This could involve developing new definitions of differentiability that account for the unique stratified structure of Carnot groups, allowing for the inclusion of functions that are piecewise smooth or exhibit singularities.
Moreover, the techniques can be adapted to explore the interplay between the algebraic properties of the Carnot groups and the geometric properties of the functions defined on them. For instance, one could investigate how the stratification of the Lie algebra influences the differentiability of functions and their intrinsic graphs, potentially leading to new results in the theory of sub-Riemannian manifolds.

Are there any connections between the intrinsic differentiability of functions and the geometry of the underlying Carnot group that are not yet fully understood?

Yes, there are several connections between the intrinsic differentiability of functions and the geometry of the underlying Carnot group that remain to be fully understood. One area of interest is the relationship between the intrinsic differentiability of functions and the geometric properties of the intrinsic cones defined in the context of Carnot groups. While the paper establishes a framework for understanding differentiability in terms of intrinsic Lipschitz continuity, the precise geometric implications of these definitions are still being explored.
Another connection lies in the role of the stratification of the Lie algebra in determining the differentiability properties of functions. The interaction between the horizontal and vertical components of the group can lead to complex behaviors that are not yet fully characterized. Understanding how these interactions affect the intrinsic differentiability of functions could yield new insights into the structure of Carnot groups and their applications.
Additionally, the implications of intrinsic differentiability on the regularity of measures induced by intrinsic graphs are not completely understood. The interplay between geometric measure theory and the differentiability of functions could lead to new results regarding the rectifiability and regularity of sets in Carnot groups, which are crucial for applications in analysis and geometry.
Overall, further research is needed to elucidate these connections, potentially leading to a richer understanding of the geometry of Carnot groups and the functions defined on them.