Volume 19, Issue 1

We classify minimal hypersurfaces in ℝ n × S m with n , m ≥ 2 which are invariant by the canonical action of O( n ) × O( m ). We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvature hypersurfaces are all unstable.

Let X be a product of r locally compact and geodesically complete Hadamard spaces. We prove that the horospheres in X centered at regular boundary points of X are Lipschitz-( r − 2)-connected. If X has finite Assouad–Nagata dimension, then using the filling construction by R. Young in [10] this gives sharp bounds on higher order Dehn functions for such horospheres. Moreover, if Γ ⊂ Is( X ) is a lattice acting cocompactly on X minus a union of disjoint horoballs, then we get a sharp bound on higher order Dehn functions for Γ . We deduce that apart from the Hilbert modular groups already considered by R. Young, every irreducible ℚ-rank one lattice acting on a product of r Riemannian symmetric spaces of the noncompact type is undistorted up to dimension r −1 and has k -th order Dehn function asymptotic to V ( k +1)/ k for all k ≤ r − 2.

We show that a unital U in PG(2, q 2 ) containing a point P , such that at least q 2 − ϵ of the secant lines through P intersect U in a Baer subline, is an ovoidal Buekenhout–Metz unital (where ϵ ≈ 2 q for q even and ϵ ≈ q 3/2 /2 for q odd).

Is it true that a convex body K being complete and reduced with respect to some gauge body C is necessarily of constant width, i.e. does it satisfy K − K = ρ ( C − C ) for some ρ > 0? We prove this implication for several cases including the following: if K is a simplex or if K contains a smooth extreme point, then the implication holds. Moreover, we derive several new results on perfect norms.

We classify all homogeneous pseudo-embeddings of the point-line geometry defined by the points and k -dimensional subspaces of PG( n , 2), and use this to study the local structure of homogeneous full projective embeddings of the dual polar space DW(2 n − 1, 2). Our investigation allows us to distinguish n possible types for such homogeneous embeddings. For each of these n types, we construct a homogeneous full projective embedding of DW(2 n − 1, 2).

We classify the rank two commutative semifields which are 8-dimensional over their center 𝔽 q . This is done using computational methods utilizing the connection to linear sets in PG(2, q 4 ). We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over 𝔽 3 . The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.

We study cohomological properties of complex manifolds. In particular, under suitable metric conditions, we extend to higher dimensions a result by A. Teleman, which provides an upper bound for the Bott– Chern cohomology in terms of Betti numbers for compact complex surfaces according to the dichotomy b 1 even or odd.

We give necessary and sufficient conditions to specify vector bundles of conformal blocks for 𝔰𝔩 2 m with rectangular weights which have ranks 0, 1, and larger than 1. We show that the first Chern classes of such rank one bundles determine a finitely generated subcone of the nef cone.

It is known that there exist hyperplane arrangements with the same underlying matroid that admit non-homotopy equivalent complement manifolds. Here we show that, in any rank, complex central hyperplane arrangements with up to 7 hyperplanes and the same underlying matroid are isotopic. In particular, the diffeomorphism type of the complement manifold and the Milnor fiber and fibration of these arrangements are combinatorially determined, that is, they depend only on the underlying matroid. To prove this, we associate to every such matroid a topological space, that we call the reduced realization space ; its connectedness, shown by means of symbolic computation, implies the desired result.

We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincaré duality for cohomology of tropical manifolds, which are polyhedral spaces locally given by Bergman fans of matroids.

Let ∆ φ = ∆ − ∇ φ ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ = e − φ dν on an n -dimensional compact Riemannian manifold ( M , g ), where g = g ( t ) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of ∆ φ and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.