Remarks on Manifolds with Two-Sided Curvature Bounds

We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

Variable Anisotropic Hardy Spaces with Variable Exponents

Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).

On Weak Super Ricci Flow through Neckpinch

In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊 n . We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.

Sub-Finsler Horofunction Boundaries of the Heisenberg Group

We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics, that is, those that arise as asymptotic cones of word metrics, on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.

Concentration of Product Spaces

We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov’s book [4]. We obtain a complete answer for this question.

Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/s)-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s. At the endpoint, α = s, a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/s)-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s-dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n, the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n.

5-Point CAT(0) Spaces after Tetsu Toyoda

We give another proof of Toyoda’s theorem that describes 5-point subspaces in CAT(0) length spaces.

On the Volume of Sections of the Cube

We study the properties of the maximal volume k-dimensional sections of the n-dimensional cube [−1, 1] n . We obtain a first order necessary condition for a k-dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝ n onto a k-dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube [−1, 1] n , n ≥ 2.

Density and Extension of Differentiable Functions on Metric Measure Spaces

We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions de-fined on these metric measure spaces.

Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds

In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.