Rosa M. B. Chaves

Björling problem for maximal surfaces in Lorentz–Minkowski space

We introduce a new approach to the local study of maximal surfaces in Lorentz–Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Bjorling-type problem in Lorentz–Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces.

The Cauchy problem for improper affine spheres and the Hessian one equation

We give a conformal representation for improper affine spheres which is used to solve the Cauchy problem for the Hessian one equation. With this representation, we characterize the geodesics of an improper affine sphere, study its symmetries and classify the helicoidal ones. Finally, we obtain the complete classification of the isolated singularities of the Hessian one Monge-Ampere equation.