Using results from non-abelian Hodge theory for klt spaces developed by Greb, Kebekus, Peternell and Taji, we deduce necessary and sufficient conditions for projective varieties with klt singularities to be uniformized by bounded symmetric domains. This generalizes a well known result of Simpson to the singular setting. We apply this to obtain explicit Miyaoka-Yau-type equalities to characterize singular quotients of the four classical irreducible bounded symmetric domains, and the polydisk.
Various results about a divisor function with Diophantine properties are obtained, including a simple asymptotic formula for its sum and a Voronoï-type formula. The proofs rely on analytic properties of certain Dirichlet series that are expressed in terms of Hecke’s zeta functions with Grössencharaktere associated to a real quadratic number field. Also used are new estimates and asymptotics for the standard hypergeometric function that are uniform in parameters, which are of independent interest.
The aim of this paper is to establish multiple positive normalized solutions (u,v,λ_1,λ_2)∈ H^1(ℝ^N,ℝ^2)×ℝ^2 to the following coupled Schrödinger system involving Sobolev critical exponent: -Δu+λ_1 u=μ_1|u|^p-2u+να|u|^α-2u|v|^β, x∈ℝ^N, -Δv+λ_2 v=μ_2|v|^q-2v+νβ|v|^β-2v|u|^α, x∈ℝ^N, ∫_ℝ^N|u|^2dx=a, ∫_ℝ^N|v|^2dx=b, N≥ 3, where μ_1,μ_2, ν, a, b>0. We are particularly interested in the mass mixed case that 2 1, β>1, and α+β=2^*:=2N/N-2. For sufficiently small ν>0, we demonstrate that the above system admits two positive solutions, one of which serves as a local minimizer, and the other as a mountain pass solution. By developing some new technical lemmas on the interaction estimates, we are managed to resolves Soave's open problem [J. Funct. Anal., 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions N≥ 3. Our results also significantly extend the result of Gou and Jeanjean [Nonlinearity, 2018, Theorem 1.1] to the Sobolev critical coupled case and removing the hypothesis “either p,q≤ α+β-2/N or |p-q|≤2/N" for N≥ 5. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter ν, and the limiting profiles for ν→ 0^+.
We provide several sufficient symmetry conditions for an embedded free boundary minimal annulus in the unit 3-ball to be congruent to the critical catenoid. First, we show that an annulus with either two symmetry planes or a symmetry plane that does not intersect the boundary must be congruent to the critical catenoid. As a corollary, any embedded free boundary minimal annulus in the half-ball satisfying certain boundary conditions is also the critical catenoid. Building on a symmetry principle, we further show that if the boundary consists of two congruent components and is invariant under reflection through a plane, then the surface is congruent to the critical catenoid.
For every stable presentably symmetric monoidal ∞ -category 𝒞 and every non-unital ∞ -operad 𝒪 in 𝒞 , where , we construct a Koszul duality adjunction TQ_𝒪: Alg_𝒪(𝒞) ⇆Coalg_𝒪^∨(𝒞): Prim_𝒪 between 𝒪 -algebras in 𝒞 and coalgebras over the Koszul dual ∞ -cooperad of 𝒪 . We prove that if all norm maps in 𝒞 associated to symmetric groups are equivalences, the unit of Koszul duality X →Prim_𝒪(TQ_𝒪(X)) identifies with the canonical map X → X^∧ := lim _n ≥ 1τ _n(𝒪) ∘ _𝒪X to the limit of the TQ_𝒪 -completion tower X ≃𝒪∘ _𝒪X → ... →τ _n(𝒪) ∘ _𝒪X → ... →τ _1(𝒪) ∘ _𝒪X= TQ_𝒪(X). We apply this result to the Koszul duality between the shifted spectral Lie ∞ -operad and the cocommutative cooperad to construct a derived enveloping Hopf algebra functor Alg_Lie(𝒞) →Hopf(𝒞) from Lie algebras in 𝒞 to cocommutative Hopf algebras in 𝒞 and deduce a derived version of the Milnor-Moore theorem: for every rational stable presentably symmetric monoidal ∞ -category 𝒞 the derived enveloping Hopf algebra functor Alg_Lie(𝒞) →Hopf(𝒞) is fully faithful.
