manuscripta mathematica

In this paper we present sufficient conditions for the non-existence of stable, complete, noncompact, constant mean curvature hypersurfaces in certain (n+1)-dimensional Riemannian manifolds.

In this article we consider nonsingular unimodular lattices. With every element, which is divisible by 2, we associate a so-called Brown invariant and prove a congruence relating the self-intersection number of the element, the signatur of the corresponding quadratic form and the Brown invariant, defined by this element. This result will be applied to study involutions on 4-dimensional manifolds.

We show that pseudoconvex domains have the Mergelyan property if the Levi form is degenerate on a sufficiently small set in the boundary. This includes the case when the weakly pseudoconvex points all lie on a smooth curve.

This paper treats the existence of positive solutions of $$-\Delta u + V(x) u = \uplambda f(u)$$ in $${\mathbb {R}}^N$$ . Here $$N \ge 1$$ , $$\uplambda > 0$$ is a parameter and f(u) satisfies conditions only in a neighborhood of $$u=0$$ . We shall show the existence of positive solutions with potential of trapping type or $${\mathcal {G}}$$ -symmetric potential where $${\mathcal {G}} \subset O(N)$$ . Our results extend previous results (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family $$(u_\uplambda )_{\uplambda \ge \uplambda _0}$$ of positive solutions as $$\uplambda \rightarrow \infty $$ .

It is shown that a mixed Ricci-flat twisted product semi-Riemannian manifold can be expressed as a warped product semi-Riemannian manifold. As a consequence, any Einstein twisted product semi-Riemannian manifold is in fact, a warped product semi-Riemannian manifold.

Gorenstein rings occur in a multitude of different guises: as rings of invariants, as coordinate rings of certain determinantal varieties and symmetric semigroup curves, as representatives of linkage classes, and so on. In an attempt to unify this motley collection of examples (at least for finite projective dimension) one seeks a generic free resolution whose specializations yield all examples of given embedding codimension. The present paper describes a resolution for codimension four, not generic, but general enough to encompass many diverse examples. The structure of this resolution is intimately related to the differential, graded, commutative algebra that it supports, and to the deformation theory of codimension four Gorenstein algebras. These ideas are brought together in the determination of the singular locus of certain codimension four Gorenstein varieties. More generally they suggest a classification of codimension four Gorenstein rings that begins to impose some order on the examples.