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Let$X$be a complex manifold with finitely many ends such that each end is either$q$-concave or ($n−q$)-convex. If $$q< \tfrac{1}{2}n$$ , then we prove that$H$^{p$n−q$}$(X)$is Hausdorff for all$p$. This is not true in general if $$q \geqslant \tfrac{1}{2}n$$ (Rossi’s example with$n$=2 and$q$=1). If all ends are$q$-concave, then this is the classical Andreotti-Vesentini separation theorem (and holds also for $$q \geqslant \tfrac{1}{2}n$$ ). Moreover the result was already known in the case when the$q$-concave ends can be ‘filled in’ (again also for $$q \geqslant \tfrac{1}{2}n$$ ). To prove the result we first have to study Serre duality for the case of more general families of supports (instead of the family of all closed sets and the family of all compact sets) which is the main part of the paper. At the end we give an application to the extensibility of CR-forms of bidegree ($p, q$) from ($n−q$)-convex boundaries, $$q< \tfrac{1}{2}n$$ .

In this paper, we classify n-dimensional (n ≥ 3) complete non-compact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact nonflat locally conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.

For almost all Riemannian metrics (in the C1 Baire sense) on a closed manifold M^{n+1}, 3 \leq n + 1 \leq 7, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of a result by Irie and the first two authors, that established density of minimal hypersurfaces for generic metrics. The main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich and the first two authors .

In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwells equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yees scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved