Band 15, Heft 4

We revisit the configuration DCD(4) of Danzer, a great inspiration for our work. This configuration of type (35 4 ) falls into an in_nite series of geometric point-line configurations DCD(n). Each DCD(n) is characterized combinatorially by having the Kronecker cover over the Odd graph On as its Levi graph. Danzer’s configuration is deeply rooted in Pascal’s Hexagrammum Mysticum. Although the combinatorial configuration is highly symmetric, we conjecture that there are no geometric point-line realizations with 7- or 5-fold rotational symmetry; on the other hand, we found a point-circle realization having the symmetry group D 7 , the dihedral group of order 14.

In this short note we show the following result: Let (M 2n+1 , g) with n ≥ 2 be a compact Sasaki manifold with positive transverse orthogonal bisectional curvature. Then π 1 (M) is finite, and the universal cover of (M 2n+1 , g) is isomorphic to a simple metric on a weighted Sasaki sphere.We also get some results in the case of nonnegative transverse orthogonal bisectional curvature under some additional conditions. This extends recent work of He and Sun. The proof uses the Sasaki-Ricci flow.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure.We call them real open book decompositions.We show that each real open book carries a real contact structure and two real contact structures supported by the same real open book decomposition are equivariantly isotopic. We also show that every real contact structure on a closed 3-dimensional real manifold is supported by a real open book. Finally, we conjecture that two real open books on a real contact manifold supporting the same real contact structure are related by positive real stabilizations and equivariant isotopy, and that the Giroux correspondence applies to real manifolds as well, namely that there is a one-to-one correspondence between the real contact structures on a real 3-manifold up to equivariant contact isotopy and the real open books up to positive real stabilization. Meanwhile, we study some examples of real open books and real Heegaard decompositions in lens spaces.

It is shown that codimension one parabolic foliations of complex manifolds are holomorphic. This is proved using the facts that codimension one foliations of complex manifolds are necessarily locally Monge- Ampère foliations and that parabolic leaves cannot have hyperbolic behavior. The result holds true also for locally Monge-Ampère foliations with parabolic leaves of arbitrary codimension.

The paper establishes a series of gradient estimates for positive solutions to the heat equation on a manifold M evolving under the Ricci flow, coupled with the harmonic map flow between M and a second manifold N. We prove Li-Yau type Harnack inequalities and we consider the cases when M is a complete manifold without boundary and when M is compact without boundary.

We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density.We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such spaces vertical fibers are isoperimetric. Our main hypothesis is that the surface area of a fiber be a convex function of the volume it bounds. We apply our results in the specific case of ℝ n − {0} realized as the warped product (0, ∞) × r S n−1 , providing many new examples of densities where spheres about the origin are isoperimetric, including simple densities with finite volume, simple densities that at the origin are neither log-convex nor smooth, and non-simple densities. We also generalize the results of Kolesnikov and Zhdanov on large balls in Rn with increasing strictly log-convex simple density. We situate our work in relation to the Log-Convex Density Conjecture of Rosales et al. and the recent work by Morgan, Ritoré, and others on formulating a generalized log-convex density/stable spheres conjecture.

We continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paperwe mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitableweights are taken into account. Finally,we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

Constructing constant width sets or, more generally, complete sets in Banach spaces seems to be a not so easy task. A new construction working in separable Banach spaces is presented, and Bavaud’s and Lachand-Robert and Oudet’s constructions of complete sets are extended to a more general context.

We determine the Euclidean distortion of finite generalized polygons by means of semidefinite programing.

In [3] Bozek introduced a class of solvable Lie groups M 2n+1 . Calvaruso, Kowalski and Marinosci in [9] have studied homogeneous geodesics on these homogeneous spaces with an arbitrary odd dimension. In [1] we have studied some other geometrical properties on these spaces with dimension five. Our aim in this paper is to extend those geometrical properties for an arbitrary odd dimension in both Riemannian and Lorentzian cases. In fact we first obtain all of the descriptions of their homogeneous Lorentzian and Riemannian structures and their types. Then we calculate the energy of an arbitrary left-invariant vector field X on these spaces and in the Lorentzian case we prove that no left-invariant unit time-like vector fields on these spaces are critical points for the space-like energy. There is also a proof of non-existence of invariant contact structures and left-invariant Ricci solitons on these homogeneous spaces.

We prove the Beckman-Quarles theorem in a uniform way for all finite dimensions n ≥ 2.