Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.

Hyperkähler isometries of K3 surfaces

We consider symmetries of K3 manifolds. Holomorphic symplectic automorphisms of K3 surfaces have been classified, and observed to be subgroups of the Mathieu group M23. More recently, automorphisms of K3 sigma models commuting with SU(2) × SU(2) R-symmetry have been classified by Gaberdiel, Hohenegger, and Volpato. These groups are all subgroups of the Conway group. We fill in a small gap in the literature and classify the possible hyperkähler isometry groups of K3 manifolds. There is an explicit list of 40 possible groups, all of which are realized in the moduli space. The groups are all subgroups of M23.

The Measure Preserving Isometry Groups of Metric Measure Spaces

Bochner's theorem says that if M is a compact Riemannian manifold with negative Ricci curvature, then the isometry group Iso(M) is finite. In this article, we show that if (X,d,m) is a compact metric measure space with synthetic negative Ricci curvature in Sturm's sense, then the measure preserving isometry group Iso(X,d,m) is finite. We also give an effective estimate on the order of the measure preserving isometry group for a compact weighted Riemannian manifold with negative Bakry-Emery Ricci curvature except for small portions.

Octonions, triality, the exceptional Lie algebra F4 and polar actions on the Cayley hyperbolic plane

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras [Formula: see text] and [Formula: see text], i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we determine all polar actions on the Cayley hyperbolic plane which leave a totally geodesic subspace invariant.

ORDINALITY OF ISOMETRY GROUPS OF HAMMING SPACES OF PERIODIC SEQUENCES

We study Hamming spaces (known also as measure algebras). For all Steinitz numbers s , we find cardinalities of the groups of isometries of Hamming spaces of s -periodic sequences and the group of automorphisms of such space and we prove that are both cardinalities equal to Pic 1.