arXiv subjects
Publications and source records attributed to Alexander Lytchak.
We prove the equivalence of several notions of infinitesimally Euclidean-like behaviour for metric surfaces appearing in the literature. We obtain in particular that metric surfaces with property (ET) are universally infinitesimally Hilbertian.
From previous results it follows that probability distributions on metric spaces featuring unique Fréchet means are dense among measures admitting means in both the quadratic Wasserstein metric and the total variation metric. Additionally, we show that the converse also holds on complete finite-dimensional noncontractible Alexandrov spaces with curvatures bounded from below, which encompass compact manifolds without boundaries and nonmanifold shape spaces: Probability distributions featuring nonunique Fréchet means are also dense in both the Wasserstein metric and the total variation metric. Moreover, in either metric, no nonempty open set of probability measures admits a continuous selection of means. This sharpens a previous result on zero reach of the Dirac embedding. Our argument combines cut-locus avoidance for atoms with a topological obstruction. For Riemannian manifolds, all of our conclusions hold for all exponents $1 < p < \infty$, also.
We characterize lower curvature bounds on metric surfaces by the Bishop--Gromov inequality for metric balls.
We prove that every closed Lipschitz curve in Euclidean space extends to a Lipschitz disc of least area, with Lipschitz constant controlled by that of the boundary curve. Our proof gives a new approach to the classical Douglas--Radó solution of the Plateau problem but accommodates arbitrary self-intersections and avoids Sobolev maps. The argument extends to metric spaces with curvature bounded above in the sense of Alexandrov.
We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.
We generalize de Rham's decomposition theorem to proper metric spaces: every proper metric space has a unique decomposition as the product of a Euclidean space and a finite or countable pointed $\ell^2$-product of nontrivial irreducible non-Euclidean factors. More generally, every complete metric space splits canonically into a Hilbert factor, a pointed $\ell^2$-product of irreducible non-Euclidean factors, and a factor with no nontrivial irreducible factors. These results partially answer a question of Ulam.
We simplify Goodwin's construction of a non-compact closed convex hull in a CAT(0) space and show that three points suffice for the example.
We prove that the covering dimension of a metric space with curvature bounded above equals its geometric dimension, without a separability assumption. This resolves a conjecture formulated by Kleiner that goes back to Gromov's work.
We prove that any convex flat subset in a complete Euclidean building is contained in an apartment of the maximal system of apartments.
We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fréchet functions and we discuss extensions of the statement to Fréchet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general metric spaces.
We prove that any complete metric space has a unique decomposition as a direct product of a possibly finite or zero-dimensional Hilbert space and a space that does not split off lines.
We prove that a Riemannian submersion between smooth, compact, non-negatively curved Riemannian manifolds has to be smooth, resolving a conjecture by Berestovskii--Guijarro. We show that without any curvature assumption, the smoothness of the base is implied by the smoothness of the total space. Results are proven in the much more general setting of submetries. These are metric generalizations of Riemannian submersions and isometric actions, which have recently appeared in different areas of geometry.
We discuss regularity statements for equidistant decompositions of Riemannian manifolds and for the corresponding quotient spaces. We show that any stratum of the quotient space has curvature locally bounded from both sides.
We show that a length-minimizing disk inherites the upper curvature bound of the target. As a consequence we prove that harmonic discs and ruled discs inherit the upper curvature bound from the ambient space.
We provide new structural results on sets of positive reach in Euclidean spaces and Riemannian manifolds.
We discuss solutions of several questions concerning the geometry of conformal planes.
We analyze weak convergence on CAT(0) spaces and the existence and properties of corresponding weak topologies.
We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Bačák. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.