The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for surfaces of this kind are obtained. As a consequence, some existence and uniqueness results for maximal Möbius strips and maximal Klein bottles with one end are proved.

We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold{E}_8$ singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's dessins d'enfants.

We first obtain an optimal norm estimate for one-parameter family of operators associated with the weighted harmonic Bergman projections on the ball. We then use this result and derive an optimal norm estimate for the weighted harmonic Bergman projections.

We show that, in the complex hyperquadric, the intersection of two real forms, which are certain totally geodesic Lagrangian submanifolds, is an antipodal set whose cardinality attains the smaller 2-number of the two real forms. As a corollary of the result, we know that any real form in the complex hyperquadric is a globally tight Lagrangian submanifold.

We study complete non-compact stable constant mean curvature hypersurfaces in a Riemannian manifold of bounded geometry, and prove that there are no nontrivial $L^2$ harmonic 1-forms on such hypersurfaces. We also show that any smooth map with finite energy from such a hypersurface to a compact manifold with non-positive sectional curvature is homotopic to constant on each compact set. In particular, we obtain some one-end theorems of complete non-compact weakly stable constant mean curvature hypersurfaces in the space forms.

In this article, we investigate the stability of leaves of minimal foliations of arbitrary codimension. We also study relations between Jacobi fields and vector fields which preserve a foliation, and we apply these results to Killing fields.

We classify all simple supermodules over the queer Lie superalgebra $\mathfrak{q}_2$ up to classification of equivalence classes of irreducible elements in a certain Euclidean ring.

Given a non-negative Hermitian form on the dual of the Lie algebra of a complex Lie group, one can associate to it a (possibly degenerate) Laplacian on the Lie group. Under Hörmander's condition on the Laplacian there exists a smooth time-dependent measure, convolution by which gives the semigroup generated by the Laplacian. Fixing a positive time, we may form the Hilbert space of holomorphic functions on the group which are square integrable with respect to this “heat kernel” measure. At the same time, under Hörmander's condition, the given Hermitian form extends to a time dependent norm on the dual of the universal enveloping algebra.

In previous work we have shown that, for each positive time, the Taylor map, which sends a holomorphic function to its set of Taylor coefficients at the identity element, is a unitary map from the previous Hilbert space of square integrable holomorphic functions onto a Hilbert space contained in the dual of the universal enveloping algebra.

The present paper is concerned with the behavior of these two families of Hilbert spaces when the Lie group is replaced by a product of complex Lie groups or by a quotient by a not necessarily normal subgroup. We obtain thereby the first example of unitarity of the Taylor map for a complex manifold which is not a Lie group. In addition, we determine the behavior of these spaces as the given Hermitian form varies.