Volume 6, Issue 4

We show that all critical points with respect to a point on a Riemannian surface lie on a subset of the cut locus which is locally a tree and has relatively few endpoints. Moreover, we offer some inequalities involving the length of the set of critical points.

We consider the gradient flow of higher order elliptic functionals of the type ∫ M 2 for maps from a compact Riemannian manifold M to ℝ n with image contained in another compact manifold, called the extrinsic polyharmonic map heat flow. We prove that smooth initial values can be continued as an eternal solution to the flow if M is of dimension < 2 m . In the critical case dim M = 2 m , we find a unique eternal weak solution which is smooth except possibly for finitely many times. A singularity can occur only if a “bubble” separates, using up a certain amount of energy.

Starting with carefully chosen sets of points in the Desarguesian affine plane AG(2; q 2 ) and using an idea first formulated by E. Shult, several infinite families of translation ovoids of the Hermitian surface are constructed. Various connections with locally Hermitian 1-spreads of 𝒬 − (5, q ) and semifield spreads of PG(3, q ) are also discussed. Finally, geometric characterization results are developed for the translation ovoids arising in the so-called classical and semiclassical settings.

Let ( M , ℰ) be a generalized polarized manifold, i.e., a pair of an n -dimensional smooth projective variety M and an ample vector bundle ℰ of rank r on M . Let τ be the nef value of a polarized manifold ( M , det ℰ), i.e., the minimum of the set of real numbers t such that K M + t det ℰ is nef; we have τ r ≤ n + 1 by Mori's theory. In this paper we classify the pairs ( M , ℰ) in the following two cases: (1) n − 2 ≤ τ r and τ ≥ 1; (2) n + 1 − τ r < τ ≤ 1.

We give a classification of pairs ( X, L ) consisting of a smooth complex n -dimensional projective variety X and an ample line bundle L on X such that Z ∈ is a Fano manifold with − K Z = (dim Z − 3) H Z for an ample and spanned line bundle H Z on Z .

Let G be a collineation group of a finite translation plane such that G is generated by perspectivities fixing a point 0. Let M be a non-abelian minimal normal subgroup of G . We show that M is uniquely determined, that the translation plane has characteristic 2, and that M ≅ L 2 ( q ), q even, or M ≅ Sz( q ).

We show that the solution of the Poisson equation with subanalytic data on a bounded subanalytic domain in the plane without isolated boundary points exists and we prove that the solution is definable in the o-minimal structure ℝ an; exp , provided that the domain has smooth boundary. Moreover, we investigate whether the solution is again subanalytic. At the end, we study polygons as examples of domains with nonsmooth boundary.

We give generalizations of Jung's theorem for the case when there are two Minkowski metrics in ℝ n one of which is used to define the diameter of a set M ∈ ℝ n , and the other to determine the radius of the Minkowski ball containing M .