Volume 19, Issue 3

We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

A new family of ellipsoids associated with a pair of convex bodies in ℝ n is introduced, which extends the classical John ellipsoids and the evolved L p John ellipsoids in a distinctive way. Analogues of John’s inclusion and Ball’s volume-ratio inequality are established for these new ellipsoids.

The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

For Kummer extensions given by y m = f ( x ), we discuss conditions for an integer to be a Weierstrass gap at a place P . In the case of fully ramified places, the conditions are necessary and sufficient. As a consequence, we extend independent results of several authors. Moreover, we show that if the Kummer extension is 𝔽 q 2 -maximal and f ( x ) ∈ 𝔽 q 2 [ x ] has at least two roots with the same multiplicity λ coprime to m , then m divides 2( q + 1). Under the extra condition that either m or the multiplicity of a third root of f ( x ) is odd, we conclude that m divides q + 1.

In previous work [15] the authors defined transforms for non-conformal harmonic maps from a Riemann surface into the 3-sphere. An observation from that study was that two invariants, a real and a complex function, determine a non-conformal harmonic map up to isometries of the 3-sphere. We now show that if the first invariant of a harmonic map and its transformed map are the same, then these maps are either congruent or the harmonic map belongs to a particular 1-parameter family. Inspired by this result we discuss the Bonnet problem for non-conformal harmonic maps: to what extent is a harmonic map determined by its first invariant?

The resolutions and maximal sets of compatible resolutions of all 2-(120,8,1) designs arising from maximal (120,8)-arcs, and the 2-(52,4,1) designs arising from previously known maximal (52,4)-arcs, as well as some newly discovered maximal (52,4)-arcs in the known projective planes of order 16, are computed. It is shown that each 2-(120,8,1) design associated with a maximal (120,8)-arc is embeddable in a unique way in a projective plane of order 16. This result suggests a possible strengthening of the Bose–Shrikhande theorem about the embeddability of the complement of a hyperoval in a projective plane of even order. The computations of the maximal sets of compatible resolutions of the 2-(52,4,1) designs associated with maximal (52,4)-arcs show that five of the known projective planes of order 16 contain maximal arcs whose associated designs are embeddable in two nonisomorphic planes of order 16.

We give a new proof of the theorem of Beauville and Voisin about the decomposition of the small diagonal of a K3 surface S . Our proof is explicit and works with the embedding of S in Pg${\mathbb P}^g$ . It is different from the one used by Beauville and Voisin, which employed the existence of one-parameters families of elliptic curves

By [4] a doubly transitive, non-solvable dimensional dual hyperoval D is isomorphic either to the Mathieu dual hyperoval or to a quotient of a Huybrechts dual hyperoval. In order to determine all doubly transitive dimensional dual hyperovals, it remains to classify the solvable ones, and this paper is a contribution to this problem. A doubly transitive, solvable dimensional dual hyperoval D of rank n is defined over 𝔽 2 and has an automorphism of the form ES , where E is elementary abelian of order 2 n and S ≤ Γ L(1, 2 n ); see Yoshiara [12]. The known examples D are bilinear. In [1] the bilinear , doubly transitive, solvable dimensional dual hyperovals D of rank n with GL(1, 2 n ) ≤ S are classified. Here we present two new classes of non - bilinear , doubly transitive dimensional dual hyperovals. We also consider universal covers of doubly transitive dimensional dual hyperovals, since they are again doubly transitive dimensional dual hyperovals by [2, Cor. 1.3]. We determine the universal covers of the presently known doubly transitive dimensional dual hyperovals.

Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X ,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

At each point in an immersed surface in ℝ 4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. Recently, at the singular point of a corank 1 singular surface in ℝ 3 , a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in ℝ 4 to ℝ 3 in a tangent direction, corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in ℝ 4 at a certain point to the geometry of the projection of the surface to ℝ 3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.

By analogy with associative and co-associative cases, we study a class of three-dimensional non-orientable submanifolds of manifolds with a G 2 -structure, modelled on planes lying in aspecial G 2 -orbit. An application of the Cartan–Kähler theory shows that some three-manifolds can be presented in this way. We also classify all the homogeneous ones in ℝℙ 7 .

The filling volume functions of the n -th quaternionic Heisenberg group grow, up to dimension n , as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n + 1, which is strictly faster than the growth rate of the ( n + 1)-dimensional filling volume function of the Euclidean space.

We exploit an ansatz in order to construct power series expansions for pairs of conjugate functions defined on domains of Euclidean 3-space. Convergence properties of the resulting series are investigated. Entire solutions which are not harmonic are found as well as a 2-parameter family of examples which contains the Hopf map.