Immersions with nowhere vanishing mean curvature vector

Mean Curvature of Riemannian Immersions

1. Let M and M' denote complete riemannian manifolds of dimension n and m respectively, and suppose that M is compact and oriented. For simplicity we assume that both manifolds and their metrics are smooth (i.e. of class C). In terms of local co-ordinates (x, x, ...,x") on M and local co-ordinates (y,y, •••,/") on M', the riemannian metrics are written ds = gtJ dx l dx, ds' = g'aP dy* dy * wher...

Conformal Immersions of Prescribed Mean Curvature in R3

We prove the existence of (branched) conformal immersions F : S → R with mean curvature H > 0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S, H and partial results for surfaces of higher genus.

Submanifolds with Parallel Mean Curvature Vector in Pinched Riemannian Manifolds

In this paper, we prove a generalized integral inequality for submanifolds with parallel mean curvature vector in an arbitrary Riemannian manifold, and from which we obtain a pinching theorem for compact oriented submanifolds with parallel mean curvature vector in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by Alencar-do Carmo and Hong-Wei Xu.

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

LOGARITHMIC CONNECTIONS WITH VANISHING p-CURVATURE

We examine logarithmic connections with vanishing p-curvature on smooth curves by studying their kernels, describing them in terms of formal local decomposition. We then apply our results in the case of connections of rank 2 on P, classifying such connections in terms of self-maps of P with prescribed ramification.