Volume 2020, Issue 764

Motivated by the notion of K{K\hskip-0.284528pt}-gentle partition of unity introduced in [J. R. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Invent. Math. 160 (2005), no. 1, 59–95] and the notion of K{K\kern-0.284528pt}-Lipschitz retract studied in [S. I. Ohta, Extending Lipschitz and Hölder maps between metric spaces, Positivity 13 (2009), no. 2, 407–425], we study a weaker notion related to the Kantorovich–Rubinstein transport distance that we call K{K\kern-0.284528pt}-random projection. We show that K{K\kern-0.284528pt}-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak*{{}^{*}} continuous operators. Finally, we use this notion to characterize the metric spaces (X,d){(X,d)} such that the free space ℱ⁢(X){\mathcal{F}(X)} has the bounded approximation propriety.

For a classical group over a non-archimedean local field of odd residual characteristic p , we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p , as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant nonnegative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifolds of nonnegative sectional curvature. Using a concept of “duality” for strictly convex hypersurfaces, we also obtain a new type of inequality, so-called “pseudo”-Harnack inequality, for expanding flows in the sphere and in the hyperbolic space.

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a 𝒞1{\mathcal{C}^{1}} dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a 𝒞1{\mathcal{C}^{1}} neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [24] which applies only to calibrated submanifolds of special holonomy ambient manifolds.

We prove that half spaces are the only stable nonlocal s -minimal cones in ℝ3{\mathbb{R}^{3}}, for s∈(0,1){s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s -minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from s=1{s=1}. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s -perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules for simply-laced Dynkin types except for type E8{\mathrm{E}_{8}}.

We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold ℳ=M×𝕊N×I{\mathcal{M}=M\times\mathbb{S}^{N}\times I}, whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for N→∞{N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on ℳ{\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle 𝒪⁢ℳ{\mathcal{OM}}. As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

Given a semisimple complex linear algebraic group G{{G}} and a lower ideal I in positive roots of G , three objects arise: the ideal arrangement 𝒜I{\mathcal{A}_{I}}, the regular nilpotent Hessenberg variety Hess⁡(N,I){\operatorname{Hess}(N,I)}, and the regular semisimple Hessenberg variety Hess⁡(S,I){\operatorname{Hess}(S,I)}. We show that a certain graded ring derived from the logarithmic derivation module of 𝒜I{\mathcal{A}_{I}} is isomorphic to H*⁢(Hess⁡(N,I)){H^{*}(\operatorname{Hess}(N,I))} and H*⁢(Hess⁡(S,I))W{H^{*}(\operatorname{Hess}(S,I))^{W}}, the invariants in H*⁢(Hess⁡(S,I)){H^{*}(\operatorname{Hess}(S,I))} under an action of the Weyl group W of G . This isomorphism is shown for general Lie type, and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G/B{G/B}. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H*⁢(G/B)→H*⁢(Hess⁡(N,I)){H^{*}(G/B)\to H^{*}(\operatorname{Hess}(N,I))} announced by Dale Peterson and an affirmative answer to a conjecture of Sommers and Tymoczko are immediate consequences. We also give an explicit ring presentation of H*⁢(Hess⁡(N,I)){H^{*}(\operatorname{Hess}(N,I))} in types B , C , and G . Such a presentation was already known in type A and when Hess⁡(N,I){\operatorname{Hess}(N,I)} is the Peterson variety. Moreover, we find the volume polynomial of Hess⁡(N,I){\operatorname{Hess}(N,I)} and see that the hard Lefschetz property and the Hodge–Riemann relations hold for Hess⁡(N,I){\operatorname{Hess}(N,I)}, despite the fact that it is a singular variety in general.

We prove a generalized version of Kazhdan’s theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence {Sn→S}{\{S_{n}\rightarrow S\}} of finite Galois covers of a hyperbolic Riemann surface S , converging to the universal cover . The theorem states that the sequence of forms on S inherited from the canonical forms on Sn{S_{n}}’s converges uniformly to (a multiple of) the hyperbolic form. We prove a generalized version of this theorem, where the universal cover is replaced with any infinite Galois cover . Along the way, we also prove a Gauss–Bonnet-type theorem in the context of arbitrary infinite Galois covers.