Communications in Contemporary Mathematics

We classify the subgroups of the automorphism group of the product of four projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi–Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is nontrivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K² = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

We give a precise formulation of T-duality for Ramond–Ramond fields. This gives a canonical isomorphism between the "geometrically invariant" subgroups of the twisted differentialK-theory groups associated to certain principal torus bundles. Our result combines topological T-duality with the Buscher rules found in physics.

We study intrinsic deformations of Levi flat structures on a smooth manifold. A Levi flat structure on a smooth manifold L is a couple (ξ, J) where ξ ⊂ T(L) is an integrable distribution of codimension 1 and J : ξ → ξ is a bundle automorphism which defines a complex integrable structure on each leaf. A deformation of a Levi flat structure (ξ, J) is a smooth family {(ξ_t, J_t)}_t∈]-ε,ε[ of Levi flat structures on L such that (ξ₀, J₀) = (ξ, J). We define a complex whose cohomology group of order 1 contains the infinitesimal deformations of a Levi flat structure. In the case of real analytic Levi flat structures, this cohomology group is where (𝒵*(L), δ, {⋅,⋅}) is the differential graded Lie algebra associated to ξ.

In this paper we consider a 2D nonlinear and nonlocal model describing the dynamics of the dislocation densities. We prove the local well-posedness of strong solution in the suitable functional framework, and we show the global well-posedness for some dissipative cases by the method of nonlocal maximum principle.

In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in ℝⁿ. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the L^p Hardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.

We provide general discussion on the lower bound of Ricci curvature along Kähler–Ricci flows over closed manifolds. The main result is the non-existence of Ricci lower bound for flows with finite time singularities and non-collapsed global volume. As an application, we give examples showing that positivity of Ricci curvature would not be preserved by Ricci flow in general.

An explicit realization of the affine Lie algebra at the critical level is constructed using a mixture of bosons and parafermions. Subsequently, a representation of the associated Lepowsky–Wilson Z-algebra is given on a space of the tensor product of bosonic fields and certain semi-infinite wedge products.