arXiv subjects

Alvaro Pampano

Publications and source records attributed to Alvaro Pampano.

20 records · Page 2Linked to original sources

arXiv2019

Totally Biharmonic Hypersurfaces in Space Forms and 3-Dimensional BCV Spaces

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi-Cartan-Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only non-trivial example of a totally biharmonic surface appears in the product space $\mathbb{S}^2(ρ)\times\mathbb{R}$.

math.DG↗

arXiv2019

Visual Curve Completion and Rotational Surfaces of Constant Negative Curvature

If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1. These geodesics can be obtained by lifting extremal curves of a total curvature type energy in the plane. We completely solve this variational problem, geometrically. Moreover, we also show a way of constructing rotational surfaces of constant negative curvature in R3 by evolving these extremal curves under their associated binormal flow with prescribed velocity. Finally, we prove that, locally, all rotational constant negative curvature surfaces of R3 are foliated by extremal curves of these energies. Therefore, we conclude that there exists a one-to-one correspondence between the sub-Riemannian geodesics used by the brain for visual curve completion and these rotational surfaces of R3.

math.DG↗