Künneth formula in Rabinowitz Floer homology

Rabinowitz Floer homology has been investigated on submanifolds of contact type. The contact condition, however, is quite restrictive. For example, a product of contact hypersurfaces is rarely of contact type. In this article, we study Rabinowitz Floer homology for product manifolds which are not necessarily of contact type. We show for a class of product manifolds that there are infinitely many leafwise intersection points by proving the Künneth formula for Rabinowitz Floer homology.     

The Künneth formula for Hilbert complexes

A variant of the Künneth formula for tensor products of Fredholm complexes of Hilbert spaces is given.     

Going-down functors, the Künneth formula, and the Baum-Connes conjecture

We study the connection between the Baum-Connes conjecture for a locally compact group G with coeefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product . The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.     

Gap theorems for Kähler-Ricci solitons

In this paper, we prove that a gradient shrinking compact K?hler-Ricci soliton cannot have too large Ricci curvature unless it is K?hler-Einstein. Received: 23 October 2007, Revised: 28 February 2008     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

Some comparison theorems for K?hler manifolds

In this work, we will verify some comparison results on K?hler manifolds. They are: complex Hessian comparison for the distance function from a closed complex submanifold of a K?hler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang (J Differ Geom 69(1):43–47, 2005).     

Comparison and vanishing theorems for Kähler manifolds

In this paper, we consider orthogonal Ricci curvature \(Ric^{\perp }\) for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kähler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kähler manifolds with \(Ric^{\perp }>0\) is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233).     

Blaschke-Süss type theorems for closed plane curves

A C¹-class of plane closed curves is considered, which contains all ovals and rosettes. This class is divided into some subclasses for which greatest lower bounds of numbers of antipodal and arc-antipodal sets are determined.     

Grün’s theorems and class groups

In this article we show how Grün’s results in group theory can be used for studying the structure of class groups in normal extensions.     

Nagata-like theorems for almost Prüfer v-multiplication domains

Let D be an integral domain and X an indeterminate over D . We show that if S is an almost splitting set of an integral domain D , then D is an APVMD if and only if both DS and DN(S) are APVMDs. We also prove that if {Dα}α∈I is a collection of quotient rings of D such that D=∩α∈IDα has finite character (that is, each nonzero d∈D is a unit in almost all Dα) and each of Dα is an APVMD, then D is an APVMD. Using these results, we give several Nagata-like theorems for APVMDs.     

Plünnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and two finite subsets A and B of K*, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao describing the sets of small doubling in a group.     

Tight Lagrangian homology spheres in compact homogeneous Kähler manifolds

For any irreducible compact homogeneous Kähler manifold, we classify the compact tight Lagrangian submanifolds which have the ?²-homology of a sphere.     

“Integer-making” theorems

Let X = {x₁, x₂,…} be a finite set and associate to every x_i a real number α_i. Let f(n) [g (n)] be the least value such that given any family $\text{F}$ of subsets of X having maximum degree n [cardinality n], one can find integers α_i, i=1,2,… so that α_i ? α_i|<1 and

$\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤?(n)}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤}\text{g(n)}$

for all $\text{E ?}\text{F}$. We prove

$\text{f(n)≤n ? 1 and g(n)≤c(n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$

.     

Szegő limit theorems

The first Szeg limit theorem has been extended by Bump–Diaconis and Tracy–Widom to limits of other minors of Toeplitz matrices. We use a more geometric method to extend their results still further. Namely, we allow more general measures and more general determinants. We also give a new extension to higher dimensions, which extends a theorem of Helson and Lowdenslager.     

Kürzeste Wege im komplexen Gebiet

Ohne Zusammenfassung