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In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.
We provide bounds on the dimension of the abnormal set for rank 2 polarizations on metabelian Lie groups, establishing the Sard property for the end-point map of such groups. We also obtain bounds for the dimension of the Goh-abnormal set for metabelian Lie groups where the codimension of the derived subgroup is at most 2, with no assumption on the rank of the polarization. We thus infer that these polarized groups, equipped with sub-Riemannian structures, satisfy the minimizing Sard property.
We show that the ultralimit of a bounded sequence of Lipschitz maps into pointed metric spaces extends naturally to $p$-bounded sequences of Sobolev maps and that this ultralimit for Sobolev maps enjoys desirable properties. We use this to prove the stability of Dehn functions under ultraconvergence of pointed length spaces, thus resolving a problem posed by several researchers in the field. As an application, we obtain a simpler proof of a recent result of Stadler--Wenger, previously proved in the locally compact case by Lytchak--Wenger, characterizing spaces of curvature bounded above by $κ$ via an isoperimetric inequality for curves.
Let X be a variety of dimension n, and let Aut(X) be its automorphism group. When X is quasi-affine, we prove that a solvable subgroup of Aut(X) that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Our main applications concern arbitrary varieties. First, every connected solvable subgroup of Aut(X) is contained in a Borel subgroup and its derived length is ≤ n+1. Second, the notion of solvable and unipotent radicals are well defined for any subgroup of Aut(X). Third, if X is quasi-affine and connected and ℬ⊂Aut(X) is a Borel subgroup of derived length n+1, then X is isomorphic to the affine n-space 𝔸^n and ℬ is conjugate to the Jonquières subgroup.
This work introduces a surface observable for nonabelian four-dimensional BF theory with a cosmological term. The surface observable yields new 2-knot invariants that may extend beyond known examples such as the Alexander invariant. By BV pushforward, the surface observable induces an electric observable in nonabelian Yang-Mills theory, offering a concrete realization of 't Hooft operators. An application to self-dual Yang-Mills theory is also discussed.
