We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when expanded in these bases. The method uses a suitable Lagrange Grassmann bundle on the product of projective spaces. This is an extension of a nonnegativity result for Lagrangian Thom polynomials obtained by the authors previously. For a fixed specialization, other special- izations of the parameter lead to upper bounds for the coefficients of the given basis. One gets also upper bounds of the coefficients from the positivity of classical Thom polynomials (of singularities of mappings), obtained previously by the last two authors.

We show an example of a non-archimedean version of the existence part of the Calabi-Yau theorem in complex geometry. Precisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.

In this paper we prove a global existence theorem, in the direction of cosmological expansion, for sufficiently small perturbations of a family of n + 1-dimensional, spatially compact spacetimes, which generalizes the k = −1 Friedmann–Lemaˆıtre-Robertson– Walker vacuum spacetime. This work extends the result from [3].The background spacetimes we consider are Lorentz cones over negative Einstein spaces of dimension n  3. We use a variant of the constant mean curvature, spatially harmonic (CMCSH) gauge introduced in [2]. An important difference from the 3+1 dimensional case is that one may have a nontrivial moduli space of negative Einstein geometries. This makes it necessary to introduce a time-dependent background metric, which is used to define the spatially harmonic coordinate system that goes into the gauge. Instead of the Bel-Robinson energy used in [3], we here use an expression analogous to the wave equation type of energy introduced in [2] for the Einstein equations in CMCSH gauge. In order to prove energy estimates, it turns out to be necessary to assume stability of the Einstein geometry. Further, for our analysis it is necessary to have a smooth moduli space. Fortunately, all known examples of negative Einstein geometries satisfy these conditions. We give examples of families of Einstein geometries which have nontrivial moduli spaces. A product construction allows one to generate new families of examples. Our results demonstrate causal geodesic completeness of the perturbed spacetimes, in the expanding direction, and show that the scale-free geometry converges toward an element in the moduli space of Einstein geometries, with a rate of decay depending on the stability properties of the Einstein geometry.

Let T  (Y, ) be a transverse knot which is the binding of some open book, (T, ), for the ambient contact manifold (Y, ). In this paper, we show that the transverse invariant bT(T ) 2[HFK(−Y,K), defined in [LOSSz09], is nonvanishing for such transverse knots. This is true regardless of whether or not  is tight. We also prove a vanishing theorem for the invariants L and T. As a corollary, we show that if (T, ) is an open book with connected binding, then the complement of T has no Giroux torsion.