A class of discriminant varieties in the conformal 3-sphere.
Harmonic morphisms and circle actions on 3- and 4-manifolds
Harmonic morphisms are considered as a natural generalization of the analytic functions of Riemann surface theory. It is shown that any closed analytic 3-manifold supporting a non-constant harmonic morphism into a Riemann surface must be a Seifert fibre space. Harmonic morphisms from a closed 4-manifold to a 3-manifold are studied. These determine a locally smooth circle action on with possible fixed points. This restricts the topology of . In all cases, a harmonic morphism from a closed...
Curvature on a graph via its geometric spectrum
We approach the problem of defining curvature on a graph by attempting to attach a ‘best-fit polytope’ to each vertex, or more precisely what we refer to as a configured star. How this should be done depends upon the global structure of the graph which is reflected in its geometric spectrum. Mean curvature is the most natural curvature that arises in this context and corresponds to local liftings of the graph into a suitable Euclidean space. We discuss some examples.
Harmonic morphisms onto Riemann surfaces and generalized analytic functions
We study harmonic morphisms from domains in and to a Riemann surface , obtaining the classification of such in terms of holomorphic mappings from a covering space of into certain Grassmannians. We show that the only non-constant submersive harmonic morphism defined on the whole of to a Riemann surface is essentially the Hopf map. Comparison is made with the theory of analytic functions. In particular we consider multiple-valued harmonic morphisms defined on domains in and...
p-harmonic maps and minimal submanifolds.
Four-dimensional Einstein metrics from biconformal deformations
Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein -manifolds. Examples of one particular family have ends which collapse asymptotically...
Bernstein Theorems for Harmonic Morphisms from R3 and S3.
The evolution of the scalar curvature of a surface to a prescribed function
We investigate the gradient flow associated to the prescribed scalar curvature problem on compact riemannian surfaces. We prove the global existence and the convergence at infinity of this flow under sufficient conditions on the prescribed function, which we suppose just continuous. In particular, this gives a uniform approach to solve the prescribed scalar curvature problem for general compact surfaces.
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