In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at
infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual
perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study. We prove that on an ${sf RCD}(K,N)$ space $({rm X},{sf d},mathcal{H}^N)$, with $Kinmathbb R$, $Ngeq 2$, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with $(N-1)$-Ahlfors regular topological boundary coinciding with the essential boundary. The proof is based on a new Deformation Lemma for sets of finite perimeter in ${sf RCD}(K,N)$ spaces $({rm X},{sf d},mathfrak m)$ and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters. The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.
In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant section
al curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity. The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $mathfrak g$ of left-invariant vector fields on a Lie group $mathbb G$ and we assume t
hat $S$ Lie generates $mathfrak g$. We say that a function $f:mathbb Gto mathbb R$ (or more generally a distribution on $mathbb G$) is $S$-polynomial if for all $Xin S$ there exists $kin mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous definition is independent on $Xin S$, they form a finite-dimensional vector space. Second, if $mathbb G$ is connected and nilpotent we show that $S$-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of $mathfrak g$ are equivalent notions.
In this paper we prove the one-dimensional Preiss theorem in the first Heisenberg group $mathbb H^1$. More precisely we show that a Radon measure $phi$ on $mathbb H^1$ with positive and finite one-density with respect to the Koranyi distance is suppo
rted on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz maps $Asubseteq mathbb Rtomathbb H^1$. The previous theorem is a consequence of a Marstrand-Mattila type rectifiability criterion, which we prove in arbitrary Carnot groups for measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if we a priori ask that the tangent planes at a point might rotate at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup has a normal complement, our criterion applies in the particular case of one-dimensional horizontal subgroups. These results are the outcome of a detailed study of a new notion of rectifiability: we say that a Radon measure on a Carnot group is $mathscr{P}_h$-rectifiable, for $hinmathbb N$, if it has positive $h$-lower density and finite $h$-upper density almost everywhere, and, at almost every point, it admits as tangent measures only (multiple of) the Haar measure of a homogeneous subgroup of Hausdorff dimension $h$. We also prove several structure properties of $mathscr{P}_h$-rectifiable measures. First, we compare $mathscr{P}_h$-rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups and we realize that it is strictly weaker than them. Furthermore, we show that a $mathscr{P}_h$-rectifiable measure has almost everywhere positive and finite $h$-density whenever the tangents admit at least one complementary subgroup.
We prove that in arbitrary Carnot groups $mathbb G$ of step 2, with a splitting $mathbb G=mathbb Wcdotmathbb L$ with $mathbb L$ one-dimensional, the graph of a continuous function $varphicolon Usubseteq mathbb Wto mathbb L$ is $C^1_{mathrm{H}}$-regul
ar precisely when $varphi$ satisfies, in the distributional sense, a Burgers type system $D^{varphi}varphi=omega$, with a continuous $omega$. We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution $varphi$ to a Burgers type system $D^{varphi}varphi=omega$, with $omega$ continuous, is actually a broad solution to $D^{varphi}varphi=omega$. As a by-product of independent interest we obtain that all the continuous distributional solutions to $D^{varphi}varphi=omega$, with $omega$ continuous, enjoy $1/2$-little Holder regularity along vertical directions.
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.
In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are $C^1_H$ regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differen
tiability by means of Holder 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.
This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a conse
quence of the following result: there exists a $C^{infty}$ hypersurface $S$ without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure $mathcal{H}^{12}$. As a consequence, we show that for every Carnot group of Hausdorff dimension 12, any Lipschitz map defined on a subset of it with values in $S$ has $mathcal{H}^{12}$-null image. In particular, we deduce that this smooth hypersurface cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension $12$. As main consequence we have that a notion of rectifiability proposed by S.Pauls is not equivalent to one proposed by B.Franchi, R.Serapioni and F.Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset $U$ of a homogeneous subgroup of Hausdorff dimension $12$ of a Carnot group, every bi-Lipschitz map $f:Uto S$ satisfies $mathcal{H}^{12}(f(U))=0$. Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all $C^{infty}$-hypersurfaces in $mathbb H^n$ with $ngeq 2$ are countably $mathbb{H}^{n-1}timesmathbb R$-rectifiabile according to Pauls definition, even with bi-Lipschitz maps.
