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We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its sub-Riemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower sub-Riemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano.
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In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is possibly, weaker than the one introduced by Franchi, Serapioni, and Serra Cassano. Namely, we consider subsets $Gamma$ that, similarly to intrinsic Lipschitz graphs, have a cone property: there exists an open dilation-invariant subset $C$ whose translations by elements in $Gamma$ dont intersect $Gamma$. However, a priori the cone $C$ may not have any horizontal directions in its interior. In every Carnot group, we prove that the reduced boundary of every finite-perimeter subset can be covered by countably many subsets that have such a cone property. The cones are related to the semigroups generated by the horizontal half-spaces determined by the normal directions. We further study the case when one can find horizontal directions in the interior of the cones, in which case we infer that finite-perimeter subsets are countably rectifiable with respect to intrinsic Lipschitz graphs. A sufficient condition for this to hold is the existence of a horizontal one-parameter subgroup that is not an abnormal curve. As an application, we verify that this property holds in every filiform group, of either first or second type.
We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabais theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $text{Isom} (X,g)to text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.
We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G, then for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they have shown that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.
Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $text{Nil}$. We show that the diffeomorphism group $text{Diff}(X)$ deformation retracts to the isometry group $text{Isom}(X)$. Combining this with earlier work by many authors, this completes the determination the homotopy type of $text{Diff}(X)$ for any compact, orientable, prime 3-manifold $X$.
We provide a Rademacher theorem for intrinsically Lipschitz functions $phi:Usubseteq mathbb Wto mathbb L$, where $U$ is a Borel set, $mathbb W$ and $mathbb L$ are complementary subgroups of a Carnot group, where we require that $mathbb L$ is a normal subgroup. Our hypotheses are satisfied for example when $mathbb W$ is a horizontal subgroup. Moreover, we provide an area formula for this class of intrinsically Lipschitz functions.
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