A sub-Riemannian endpoint map with no regular values
We construct an endpoint map with no regular values, disproving the sub-Riemannian Sard conjecture. Remarkably, in our construction the minimizing Sard property remains valid.
math.DG↗
arXiv subjects
Luca Rizzi
Publications and source records attributed to Luca Rizzi.
Branching of regular abnormal geodesics in sub-Riemannian manifolds of rank two
We study branching of regular abnormal geodesics in sub-Riemannian manifolds of rank two. As an application, we provide examples of two new phenomena: continuous families of strictly abnormal branching geodesics, and geodesics with an accumulation of countably many branching points.
math.DG↗
Sard property for rank 2 polarizations in metabelian Lie groups
We provide sharp 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. The proof is based on a novel approach that makes essential use of tools from tame geometry. 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.
math.DG↗
Linear quadratic optimal transport and interpolation inequalities
This paper investigates the optimal transport problem within the framework of Linear Quadratic optimal control systems. We establish the well-posedness of the Monge problem and analyze the regularity of the resulting optimal transport map, extending the results obtained in [Hindawi, Pomet, Rifford, 2011] for non-negative costs. Furthermore, we study the displacement interpolation of measures and derive general interpolation inequalities for entropy functionals. Our analysis is motivated by the role of these systems as natural model spaces for comparison theory in sub-Riemannian geometry.
math.OC↗
Deformations of maps with fixed target and curvature of direct image bundles
Consider a smooth proper holomorphic fibration of complex manifolds. It is known that the semipositivity of the curvature of the direct image of the relative canonical bundle can be read in terms of the cup product with the Kodaira-Spencer class of the fibers. Motivated by this result, in this paper we study the relation between deformations of maps with fixed target and the curvature of certain direct image bundles. As a result, we find some curvature formulas and use them to prove a seminegativity result for a vector bundle of relative forms naturally related to the deformation data.
math.AG↗
On supported deformations and birational isotriviality
It is well known that the general fibers of a fibration $f\colon X\to B$ are isomorphic if the general Kodaira-Spencer class vanishes. In this paper we consider the birational analogue when the general Kodaira-Spencer class is supported on a divisor.
math.AG↗
Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures
We prove that the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) can fail in sub-Riemannian geometry. This may happen beyond step two, if the distance function is not Lipschitz in charts, and it already occurs in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results, of independent interest, on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. As an application, we find sub-Riemannian structures with pre-medium-fat distribution that do not satisfy the MCP, answering a question raised in [L. Rifford, J. \'Ec. polytech. Math. 2023]. Finally, and quite unexpectedly, we show that ideal sub-Riemannian structures can fail the MCP, and this actually happens generically for rank greater than 3 and high dimension.
math.DG↗
Curvature and relative volume forms
Using metric techniques introduced by Berndtsson, we show a result on constancy of families dominated by a constant variety and, on the opposite side, a result on the strong non isotriviality of certain families of surfaces with positive index. We also give metric interpretations of liftability of relative volume forms and of strong non isotriviality in terms of the complex conjugate of a suitable representative of the Kodaira-Spencer class.
math.AG↗
Quantitative approximate definable choices
In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite-dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest.
math.AG↗
Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups
The purpose of the paper is threefold: first, we prove optimal regularity results for the distance from $C^k$ submanifolds of general rank-varying sub-Riemannian structures. Then, we study the asymptotics of the volume of tubular neighbourhoods around such submanifolds. Finally, for the case of curves in the Heisenberg groups, we prove a Weyl's invariance result: the volume of small tubes around a curve does not depend on the way the curve is isometrically embedded, but only on its Reeb angle. The proof does not need the computation of the actual volume of the tube, and it is new even for the three-dimensional Heisenberg group.
math.DG↗
Sard properties for polynomial maps in infinite dimension
Sard's theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. It is well-known, however, that when the domain is infinite dimensional and the range is finite dimensional, the result is not true -- even under the assumption that the map is ``polynomial'' -- and a general theory is still lacking. Addressing this issue, in this paper, we provide sharp quantitative criteria for the validity of Sard's theorem in this setting. Our motivation comes from sub-Riemannian geometry and, as an application of our results, we prove the sub-Riemannian Sard conjecture for the restriction of the Endpoint map of Carnot groups to the set of piece-wise real-analytic controls with large enough radius of convergence, and the strong Sard conjecture for the restriction to the set of piece-wise entire controls.
math.DG↗
Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach
Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of \emph{contact} Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from [Etnyre, Komendarczyk, Massot - Invent. Math. 2012 and Trans. Amer. Math. Soc. 2016], and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan--Hadamard theorem.
math.DG↗
The Future of Astronomical Data Infrastructure: Meeting Report
The astronomical community is grappling with the increasing volume and complexity of data produced by modern telescopes, due to difficulties in reducing, accessing, analyzing, and combining archives of data. To address this challenge, we propose the establishment of a coordinating body, an "entity," with the specific mission of enhancing the interoperability, archiving, distribution, and production of both astronomical data and software. This report is the culmination of a workshop held in February 2023 on the Future of Astronomical Data Infrastructure. Attended by 70 scientists and software professionals from ground-based and space-based missions and archives spanning the entire spectrum of astronomical research, the group deliberated on the prevailing state of software and data infrastructure in astronomy, identified pressing issues, and explored potential solutions. In this report, we describe the ecosystem of astronomical data, its existing flaws, and the many gaps, duplication, inconsistencies, barriers to access, drags on productivity, missed opportunities, and risks to the long-term integrity of essential data sets. We also highlight the successes and failures in a set of deep dives into several different illustrative components of the ecosystem, included as an appendix.
astro-ph.IM↗
Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-\'Emery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry-\'Emery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0,N) condition, yielding a counterexample to gluing theorems in the RCD setting.
math.DG↗
Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures
Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework of synthetic Ricci curvature lower bounds. With the aim of achieving such a unification program, in this paper we initiate the study of gauge metric measure spaces.
math.DG↗
Global Kodaira Spencer class and Massey products
We define a new notion of supported global deformation class for a semistable family of complex varieties over a curve $f\colon X\to B$. We use this notion to study when $X$, possibly up to a finite covering, has a generically finite morphism onto a product $B\times Y$ with $Y$ of general type.
math.AG↗
Massey products and Fujita decomposition over higher dimensional base
Let $f\colon X\to Y$ be a semistable fibration between smooth complex varieties of dimension $n$ and $m$. This paper contains an analysis of the local systems of de Rham closed relative one forms and top forms on the fibers. In particular the latter recovers the local system of the second Fujita decomposition of $f_*\omega_{X/Y}$ over higher dimensional base. The so called theory of Massey products allows, under natural Castelnuovo-type hypothesis, to study the finiteness of the associated monodromy representations. Motivated by this result, we also make precise the close relation between Massey products and Castelnuovo-de Franchis type theorems.
math.AG↗
Cosmicflows-4
With Cosmicflows-4, distances are compiled for 55,877 galaxies gathered into 38,065 groups. Eight methodologies are employed, with the largest numbers coming from the correlations between the photometric and kinematic properties of spiral galaxies (TF) and elliptical galaxies (FP). Supernovae that arise from degenerate progenitors (type Ia Sne) are an important overlapping component. Smaller contributions come from distance estimates from the surface brightness fluctuations of elliptical galaxies and the luminosities and expansion rates of core collapse supernovae (SNII). Cepheid period-luminosity relation and tip of the red giant branch observations founded on local stellar parallax measurements along with the geometric maser distance to NGC 4258 provide the absolute scaling of distances. The assembly of galaxies into groups is an important feature of the study in facilitating overlaps between methodologies. Merging between multiple contributions within a methodology and between methodologies is carried out with Bayesian Markov chain Monte Carlo procedures. The final assembly of distances is compatible with a value of the Hubble constant of $H_0=74.6$ km s$^{-1}$ Mpc$^{-1}$ with the small statistical error of $\pm 0.8$ km s$^{-1}$ Mpc$^{-1}$ but a large potential systematic error of ~3 km s$^{-1}$ Mpc$^{-1}$. Peculiar velocities can be inferred from the measured distances. The interpretation of the field of peculiar velocities is complex because of large errors on individual components and invites analyses beyond the scope of this study.
astro-ph.CO↗