arXiv subjects

Jose A. Galvez

Publications and source records attributed to Jose A. Galvez.

21 records · Page 2Linked to original sources

arXiv2004

The space of solutions to the Hessian one equation in the finitely punctured plane

We construct the space of solutions to the elliptic Monge-Ampere equation det(D^2 u)=1 in the plane R^2 with n points removed. We show that, modulo equiaffine transformations and for n>1, this space can be seen as an open subset of R^{3n-4}, where the coordinates are described by the conformal equivalence classes of once punctured bounded domains in the complex plane of connectivity n-1. This approach actually provides a constructive procedure that recovers all such solutions to the Monge-Ampere equation, and generalizes a theorem by K. Jorgens.

math.AP↗

arXiv2003

The Cauchy problem for Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for Liouville elliptic equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the only mean curvature one surface in H3 that passes through a given curve with given unit normal along it, and provide diverse applications. In particular, topics like period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. of surfaces are treated.

math.DG↗

arXiv2003

Isometric immersions of R^2 into R^4 and pertubation of Hopf tori

We produce a new general family of flat tori in R^4, the first one since Bianchi's classical works in the 19th century. To construct these flat tori, obtained via small perturbation of certain Hopf tori in S^3, we first present a global description of all isometric immersions of R^2 into R^4 with flat normal bundle.

math.DG↗