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We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold$N$of ℂ^{$n$}to complex transversal wedges with edges containing$N$. We show that given any$v$∈ℂ^{$n$}∖($T$_{$p$}$N$+$iT$_{$p$}$N$), there exists a wedge of direction$v$whose edge contains a neighborhood of$p$in$N$, such that any continuous CR function defined locally near$p$extends holomorphically to that wedge.

We show that a properly immersed minimal hypersurface in M × R+ equals some M×{c} when M is a complete, recurrent ndimensional Riemannian manifold with bounded curvature. If on the other hand,M is not necessarily recurrent but has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph over M.

We show that any star-shaped convex hypersurface with constantWeingarten curvature in the deSitter-Schwarzschildmanifold is a sphere of symmetry.Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of the enclosed volume, and we completely describe the isoperimetric surfaces for very large enclosed volume. This complements work in H. Bray’s thesis, where isoperimetric surfaces homologous to the horizon are studied.

We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [..] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [..] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.