For a genus-1 1-bridge knot in $S^3$, that is, a (1,1)-knot, a middle tunnel is a tunnel that is not an upper or lower tunnel for some (1,1)-position. Most torus knots have a middle tunnel, and non-torus-knot examples were obtained by Goda, Hayashi, and Ishihara. We generalize their construction and calculate the slope invariants for the resulting middle tunnels. In particular, we obtain the slope sequence of the original example of Goda, Hayashi, and Ishihara.

We generalize the concept of Sato Grassmannians of locally linearly compact topological vector spaces (Tate spaces) to the Beilinson category of the “locally compact objects”, or Generalized Tate Spaces, of an exact category. This allows us to extend the Kapranov dimensional torsor Dim and determinantal gerbe Det to generalized Tate spaces and unify their treatment in the determinantal torsor. We then introduce a class of exact categories, that we call partially abelian exact, and prove that if the base category is so, then Dim and Det are multiplicative in admissible short exact sequences of generalized Tate spaces. We then give a cohomological interpretation of these results in terms of the Waldhausen K-theoretical space of the Beilinson category. Our approach can be iterated and we define analogous concepts for the successive categories of $n$-dimensional (generalized) Tate spaces. In particular we show that the category of Tate spaces is partially abelian exact, so we can extend the results for Dim and Det obtained for 1-Tate spaces to 2-Tate spaces, and provide a new interpretation in the context of algebraic $K$-theory of results of Kapranov, Arkhipov-Kremnizer and Frenkel-Zhu.

We study Tauberian conditions for the existence of Cesàro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Cesàro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee $(\mathrm{C},\beta)$ summability for a given order $\beta$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Cesàro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szàsz for Dirichlet series.

The conjecture of Blair says that there are no nonflat Riemannian metrics of nonpositive curvature associated with a contact structure. We prove this conjecture for a certain class of contact structures on closed 3-dimensional manifolds and construct a local counterexample.

Given a harmonic measure $m$ of a hyperbolic lamination $\mathcal L$ on a compact metric space $M$, a positive harmonic function $h$ on the universal cover of a typical leaf is defined in such a way that the measure $m$ is described in terms of these functions $h$ on various leaves. We discuss some properties of the function $h$. We show that if $m$ is ergodic and not completely invariant, then $h$ is typically unbounded and is induced by a probability $\mu$ of the sphere at infinity which is singular to the Lebesgue measure. A harmonic measure is called Type I (resp. Type II) if for any typical leaf, the measure $\mu$ is a point mass (resp. of full support). We show that any ergodic harmonic measure is either of type I or type II.

Let $S$ be a complex nonsingular minimal projective surface of general type with $q(S)=2$, and let $G$ be the group of the automorphisms of $S$ acting trivially on $H^2(S, \boldsybmol{Q})$. In this note we classify explicitly pairs $(S, G)$ with $G$ of order four.

We show that every reflective submanifold of a symmetric $R$-space is (geodesically) convex.