Periodic Reporting for period 4 - GeoMeG (Geometry of Metric groups)
Reporting period: 2022-03-01 to 2023-07-31
The overarching objective of this project has been to advance geometric measure theory within non-Riemannian contexts. Specifically, we harnessed Lie-group techniques to address problems in metric geometry.
Our endeavors encompassed a broad spectrum of activities, yet they shared a common thread: the exploration of subRiemannian geometry within Lie groups. This subject inherently possesses interdisciplinary qualities, as these distinct metric spaces manifest in various mathematical domains. They find application in fields such as control theory, harmonic and complex analysis, asymptotic geometry, subelliptic partial differential equations, and geometric group theory.
Our approach has revolved around investigating the geometric characteristics of metric groups and their implications for control systems and nilpotent groups. In particular, we aimed to exploit the intricate relationship between the regularity of distinguished curves, sets, and maps in subRiemannian groups, volume behavior in nilpotent groups, and embedding results.
Given that the metrics under consideration differ from the classical Riemannian metrics, we have needed to develop new methodologies and tools. The novelty of our approach lies in the fusion of Metric Geometry, Control Theory, and Lie Group Theory.
Upon concluding the project, several noteworthy achievements stand out. Firstly, we established the virtual commutativity of nilpotent groups that bilipshitzly embed into the space of integrable real functions. Secondly, we introduced the algebraic concept of the real-shadow of a Lie group and demonstrated that only groups sharing the same real-shadow can be isometric. Thirdly, we attained a multitude of results concerning the regularity of geodesic curves, sets with finite perimeter, and metrically distinguished maps.
Our endeavors encompassed a broad spectrum of activities, yet they shared a common thread: the exploration of subRiemannian geometry within Lie groups. This subject inherently possesses interdisciplinary qualities, as these distinct metric spaces manifest in various mathematical domains. They find application in fields such as control theory, harmonic and complex analysis, asymptotic geometry, subelliptic partial differential equations, and geometric group theory.
Our approach has revolved around investigating the geometric characteristics of metric groups and their implications for control systems and nilpotent groups. In particular, we aimed to exploit the intricate relationship between the regularity of distinguished curves, sets, and maps in subRiemannian groups, volume behavior in nilpotent groups, and embedding results.
Given that the metrics under consideration differ from the classical Riemannian metrics, we have needed to develop new methodologies and tools. The novelty of our approach lies in the fusion of Metric Geometry, Control Theory, and Lie Group Theory.
Upon concluding the project, several noteworthy achievements stand out. Firstly, we established the virtual commutativity of nilpotent groups that bilipshitzly embed into the space of integrable real functions. Secondly, we introduced the algebraic concept of the real-shadow of a Lie group and demonstrated that only groups sharing the same real-shadow can be isometric. Thirdly, we attained a multitude of results concerning the regularity of geodesic curves, sets with finite perimeter, and metrically distinguished maps.
We have made significant use of Lie techniques in our non-smooth geometries. These techniques encompassed various aspects, such as utilizing the theory of free nilpotent groups and algebraic quotients to simplify metric properties. We employed differential calculus on Lie groups for metric calculations related to curve shortening, semisubgroups of Lie groups, and Lie algebra classifications. Metric geometry played a crucial role. We utilized submetries, Gromov-Hausdorff limits of geodesics and distances, and visual distances. Furthermore, control theory was instrumental in our investigations.
Our project made significant strides in addressing the problem of rectifiability of finite-perimeter sets and embeddings. While the debate continues regarding the strictest notion of intrinsic rectifiability, we established a covering-type result for sets with finite subRiemannian perimeter. This coverage was achieved through sets exhibiting a cone property, akin to classical Euclidean spaces. Notably, our work challenged the expectation of achieving stronger rectifiability through intrinsically differentiable hypersurfaces, owing to the construction of pathological examples by the PI and the research team. Importantly, the PI made substantial contributions to the theory of sets with constant normal (and, more broadly, monotone sets), finally determining whether certain non-Abelian Carnot groups can be biLipschitz embedded into the space of integrable real functions.
Regarding the growth of nilpotent groups and the shape of spheres in Carnot groups, the project made progress in all planned directions. The PI derived a sharper (though potentially not the sharpest) asymptotic expansion for the volume growth of Lie nilpotent groups equipped with general subFinsler metrics. The rectifiability of boundaries of metric balls in Carnot groups was established in groups of step 2, but there is growing skepticism regarding its validity in groups of step 3.
The PI provided a comprehensive answer to which homogeneous groups satisfy the Besicovitch Covering Property. However, the general case for arbitrary Lie groups remains unsolved. Our research on abnormal curves and the regularity of geodesics yielded several results, yet many questions remain unanswered. Currently, the strongest result says that tangents of subFinsler geodesics are geodesic when projected one step lower. While the outcome eliminates many pathologies via iteration, it doesn't provide new equations for subRiemannian geodesics nor does it establish their differentiability. Progress was made on the Sard Conjecture, particularly in groups of low step.
One of the most successful aspects of the project involved determining the algebraic properties of isometries between metric groups. The description of these properties was technical and resulted in a lengthy publication. However, it comprehensively elucidated the conditions under which two groups can be isometric, namely when they share the same real-shadow. We also achieved metric classifications of 3D groups and Heintze groups in low dimensions.
Another fruitful line of research focused on the conformal equivalence of visual metrics in strictly pseudoconvex domains. Inspired by Mostow's proof of his rigidity theorem, we offered an alternative proof of a result by Fefferman on smooth extensions of biholomorphic mappings between strictly pseudoconvex domains, showing that such extensions are conformal maps between subRiemannian manifolds.
Significant progress was also made in studying metrically homogeneous spaces. We achieved several characterizations of subFinsler geometries and metric Lie groups, including a quasi-Möbius characterization of invertible homogeneous metric spaces. A comprehensive description was provided for metric Lie groups that admit dilations, shedding light on locally compact homogeneous metric spaces that also admit dilations. Our work culminated in the classification known as the 'Cornucopia of Carnot Groups in Low Dimensions,' serving as a valuable resource for researchers exploring nilpotent and Carnot groups.
Our project made significant strides in addressing the problem of rectifiability of finite-perimeter sets and embeddings. While the debate continues regarding the strictest notion of intrinsic rectifiability, we established a covering-type result for sets with finite subRiemannian perimeter. This coverage was achieved through sets exhibiting a cone property, akin to classical Euclidean spaces. Notably, our work challenged the expectation of achieving stronger rectifiability through intrinsically differentiable hypersurfaces, owing to the construction of pathological examples by the PI and the research team. Importantly, the PI made substantial contributions to the theory of sets with constant normal (and, more broadly, monotone sets), finally determining whether certain non-Abelian Carnot groups can be biLipschitz embedded into the space of integrable real functions.
Regarding the growth of nilpotent groups and the shape of spheres in Carnot groups, the project made progress in all planned directions. The PI derived a sharper (though potentially not the sharpest) asymptotic expansion for the volume growth of Lie nilpotent groups equipped with general subFinsler metrics. The rectifiability of boundaries of metric balls in Carnot groups was established in groups of step 2, but there is growing skepticism regarding its validity in groups of step 3.
The PI provided a comprehensive answer to which homogeneous groups satisfy the Besicovitch Covering Property. However, the general case for arbitrary Lie groups remains unsolved. Our research on abnormal curves and the regularity of geodesics yielded several results, yet many questions remain unanswered. Currently, the strongest result says that tangents of subFinsler geodesics are geodesic when projected one step lower. While the outcome eliminates many pathologies via iteration, it doesn't provide new equations for subRiemannian geodesics nor does it establish their differentiability. Progress was made on the Sard Conjecture, particularly in groups of low step.
One of the most successful aspects of the project involved determining the algebraic properties of isometries between metric groups. The description of these properties was technical and resulted in a lengthy publication. However, it comprehensively elucidated the conditions under which two groups can be isometric, namely when they share the same real-shadow. We also achieved metric classifications of 3D groups and Heintze groups in low dimensions.
Another fruitful line of research focused on the conformal equivalence of visual metrics in strictly pseudoconvex domains. Inspired by Mostow's proof of his rigidity theorem, we offered an alternative proof of a result by Fefferman on smooth extensions of biholomorphic mappings between strictly pseudoconvex domains, showing that such extensions are conformal maps between subRiemannian manifolds.
Significant progress was also made in studying metrically homogeneous spaces. We achieved several characterizations of subFinsler geometries and metric Lie groups, including a quasi-Möbius characterization of invertible homogeneous metric spaces. A comprehensive description was provided for metric Lie groups that admit dilations, shedding light on locally compact homogeneous metric spaces that also admit dilations. Our work culminated in the classification known as the 'Cornucopia of Carnot Groups in Low Dimensions,' serving as a valuable resource for researchers exploring nilpotent and Carnot groups.
We anticipate further progress in our ongoing tasks. For instance, we are likely to provide insights into the intriguing question of whether every subRiemannian finite-perimeter set possesses intrinsic rectifiability. Additionally, we have expectations of deriving multiple results concerning the volume growth of nilpotent groups and exploring various aspects of geodesic regularity.
Continuing to delve into the differentiability of geodesics and the subRiemannian Sard property remains an objective worth pursuing, as it leads to distinct avenues of research and has the potential to yield significant insights.
It's worth noting that our journey of partial progress has, in turn, raised new questions, some of which will be the focus of in-depth investigation, while others will be deferred to future projects.
Among the upcoming tasks on our agenda are:
. Examination of infinite geodesics in subRiemannian spaces originating from mechanics.
. Investigation into the curve shortening flow for curves within subRiemannian spaces.
. Application of the curve shortening flow in constrained dynamical scenarios through numerical methods.
. Exploration of higher-order open mapping theorems and their relevance to the end-point map.
. Deduction of formulas for subRiemannian abnormal minimizers.
. Development of a comprehensive theory for sub-Riemannian shrinking flows applied to submanifolds.
. Creation of a theory tailored to Reifenberg vanishing flat metric spaces, where the tangents are not necessarily Euclidean.
Continuing to delve into the differentiability of geodesics and the subRiemannian Sard property remains an objective worth pursuing, as it leads to distinct avenues of research and has the potential to yield significant insights.
It's worth noting that our journey of partial progress has, in turn, raised new questions, some of which will be the focus of in-depth investigation, while others will be deferred to future projects.
Among the upcoming tasks on our agenda are:
. Examination of infinite geodesics in subRiemannian spaces originating from mechanics.
. Investigation into the curve shortening flow for curves within subRiemannian spaces.
. Application of the curve shortening flow in constrained dynamical scenarios through numerical methods.
. Exploration of higher-order open mapping theorems and their relevance to the end-point map.
. Deduction of formulas for subRiemannian abnormal minimizers.
. Development of a comprehensive theory for sub-Riemannian shrinking flows applied to submanifolds.
. Creation of a theory tailored to Reifenberg vanishing flat metric spaces, where the tangents are not necessarily Euclidean.