Mathematical Images - David Brander

Some Mathematical Images - David Brander

Spherical Surfaces

Surfaces with constant positive Gauss curvature. These surfaces are produced using loop group methods, described here: Spherical surfaces

Symmetric

More examples

Pseudospherical Surfaces with singularities

Surfaces with constant negative Gauss curvature and prescribed singularities. These surfaces are produced by solving the singular geometric Cauchy problem.

Set 1

Degenerate

Planar curves

Parameter lines

Willmore Surfaces

Willmore surfaces are local minimizers of the Willmore energy $\int_S H^2 dA$, where $H$ is the mean curvature. This energy minimizing property makes the surfaces appear very smooth and natural. The images here are produced using an implementation of our solution of a generalization of Björling's problem to these surfaces. Read more...

Equivariant

Rotational

Profiles

More Images

Pseudospherical Surfaces

Surfaces with constant negative Gauss curvature. The surfaces shown here each contain a well-known curve as a principal geodesic curve. They are produced using the solution of the geometric Cauchy problem given by M. Svensson and myself in J. Differential Geom. 93, (2013), 37-66. Read more...

Parabola

Ellipse

Lemniscate

Cubic

CMC Deformations of Minimal Surfaces

These images show some examples of non-minimal CMC surfaces produced from famous minimal surfaces, using a natural deformation given in this article , which is joint work with J.F. Dorfmeister. Read more...

Catenoid

Helicoid

Enneper Surface

Scherk Surface

Singularities of Surfaces

Generic, or stable, singularities are those which persist under small local deformations of the surface through surfaces of the same class. Some of my work has been on the singularities of CMC surfaces in $2+1$-dimensional space-time.

Generic

Branch Points

Spacelike CMC

Surfaces foliated by Euler elastica

The elastica are curves that minimize bending energy. These were studied and classified by Bernoulli and Euler. Here are some images of surfaces swept out in space by (varying) elastica.

All Euler Elastica

Roughly rotational