Spherical Surfaces

We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and use these solutions to compute several new examples. We give the criteria on the geometric Cauchy data for the generic singularities, as well as for the cuspidal beaks and cuspidal butterfly singularities. We consider the bifurcations of generic one parameter families of spherical fronts and provide evidence that suggests that these are the cuspidal beaks, cuspidal butterfly and one other singularity. We also give the loop group potentials for spherical surfaces with finite order rotational symmetries and for surfaces with embedded isolated singularities.Comment: 23 pages, 18 figures. Version 3: Typos correcte

Pseudospherical surfaces with singularities

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all non-degenerate singularities of both types, as well as many degenerate singularities. We also give a method for solving the singular geometric Cauchy problem: construct a pseudospherical frontal containing a given regular space curve as a non-degenerate singular curve. The solution is unique for most curves, but for some curves there are infinitely many solutions, and this is encoded in the curvature and torsion of the curve.Comment: 26 pages, 11 figures. Version 3: examples added (new Section 6). Introduction section revise

Families of spherical surfaces and harmonic maps

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A-determined map-germs from the plane to the plane can be represented by harmonic maps.Comment: 30 pages, 7 figures. Version 2: substantial revision compared with version 1. The results are essentially the same, but some of the arguments are improved or correcte

The Benefits to People of Expanding Marine Protected Areas

This study focuses on how the economic value of marine ecosystem services to people and communities is expected to change with the expansion of marine protected areas (MPAs). It is recognised, however, that instrumental economic value derived from ecosystem services is only one component of the overall value of the marine environment and that the intrinsic value of nature also provides an argument for the conservation of the marine habitats and biodiversity

Remarks on the boundary curve of a constant mean curvature topological disc

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a condition for the surface to be totally umbilic in terms of the normal curvature.Comment: 6 pages, 1 figure. Version 2: comments and references adde

Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Bj\"orling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the partial derivatives $f_x$ and $f_y$ are given by \frac{\dd f_0}{\dd x} and $v$ along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in $L^3$. We use this to find the Bj\"orling data -- and holomorphic potentials -- which characterize cuspidal edge, swallowtail and cross cap singularities.Comment: 28 pages, 2 figures. Version 2: Figure 2 adde