Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential of this map. Considering planar sections containing the normal field, we also define normal curvatures at each point of the surface, and with respect to each tangent direction. We investigate the relations between these curvature types. Further on we prove that, under an additional hypothesis, a compact, connected surface without boundary whose Minkowski Gaussian curvature is constant must be a Minkowski sphere. Since existing literature on the subject of our paper is widely scattered, in the introductory part also a survey of related results is given.

We show that certain Fano eightfolds (obtained as hyperplane sections of an orthogonal Grassmannian, and studied by Ito–Miura–Okawa–Ueda and by Fatighenti–Mongardi) have a multiplicative Chow–Künneth decomposition. As a corollary, the Chow ring of these eightfolds behaves like that of K3 surfaces.

In this paper, we sketch some constructions providing a link between complex-analytic geometry and nonstandard algebraic geometry via a categorical point of view. The analytic category is seen as a completed fiber of a family of nonstandard algebraic geometries by applying a standard part functor. We indicate how various notions of analytic objects fit into this context (as for example banachanalytic spaces, Kähler spaces etc.)

Let p be an odd prime number. We construct explicit uniformizers for the totally ramified extension ({mathbb {Q}}_p(zeta _{p^2},root p of {p})) of the field of p-adic numbers ({mathbb {Q}}_p), where (zeta _{p^2}) is a primitive (p^2)-th root of unity.

For an arbitrary real connected Lie group G we define (mathrm {p}(G)) as the maximal integer p such that (mathbb {Z}^p) is isomorphic to a discrete subgroup of G and (mathrm {q}(G)) is the maximal integer q such that (mathbb {R}^q) is isomorphic to a closed subgroup of G. The aim of this paper is to investigate properties of these two invariants. We will show that if G is a noncompact connected Lie group, then (1le mathrm {q}(G)le mathrm {p}(G)le dim (G/K)) where K is a maximal compact subgroup of G. In the cases when G is an exponential Lie group or G is a connected nilpotent Lie group, we give explicit relationships between these two invariants and a well known Lie algebra invariant (mathcal M(mathfrak {g})), i.e. the maximum among the dimensions of abelian subalgebras of the Lie algebra (mathfrak {g}:=mathrm {Lie}(G)).

A list of possible holonomy groups with indecomposable holonomy representation contained in the exceptional, non-compact Lie group ({mathrm {G}}_2^{*}) was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1, 2 or 3, respectively. It was also shown by Fino and Kath that all algebras of Type I, and by the author that all of Type III are indeed realizable as holonomy algebras by metrics with signature (4, 3). This article proves that this is also true for all Type II algebras. Thus, there exists a realization by a metric for all indecomposable holonomy groups contained in ({mathrm {G}}_2^{*}).

Completely replicable functions play an important role in number theory and finite group theory, in particular the Monstrous Moonshine. In this paper, we give a characterization of completely replicable functions by certain symmetries.

A scale Hilbert space is a natural generalization of a Hilbert space which considers not only a single Hilbert space but a nested sequence of subspaces. Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of smooth structures in infinite dimensions. In this paper, we prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. We also give a complete description of the local invariant introduced by U. Frauenfelder for these spaces. Product mapping spaces and relative mapping spaces are also studied. Our approach is based on the spectral resolution of Laplace type operators together with the eigenvalue growth estimate.

We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module A has order p or 2p, where p is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components (F_{gamma }) of the vector-valued modular form are antisymmetric in the sense that (F_{gamma } = -F_{-gamma }) for all (gamma in A). As an application, we compute restrictions of Doi–Naganuma lifts of odd weight to components of Hirzebruch–Zagier curves.