My research interests lie within Classical Differential Geometry, specifically the Differential Geometry of Curves and Surfaces, following the foundational approach of authors such as Manfredo do Carmo (Differential Geometry of Curves and Surfaces) and Sebastián Montiel and Antonio Ros (Curves and Surfaces). In addition to this broad area, my work incorporates techniques and studies from Geometric Analysis, Calculus of Variations, and Partial Differential Equations.
A central focus of my work stems from my Ph.D. Thesis (supervised by
Sebastián Montiel), which explores constant mean curvature
surfaces in Euclidean space with a prescribed boundary. This
doctoral research serves as the foundation of my career and remains a constant motivation for
investigating new problems in differential geometry.
Some of the Research Topics:
- Constant mean curvature surfaces with prescribed boundary.
- The constant mean curvature equation: Dirichlet problem, convexity of solutions.
- Cyclic surfaces (surfaces foliated by circles) with constant curvature .
- Weingarten surfaces.
- Constant angle surfaces.
- Translation surfaces with constant curvature.
- Constant mean curvature surfaces in the steady state space.
- Minimal surfaces via the Björling problem.
- Translating solitons of the mean curvature flow.
- Minimal singular surfaces.
- Surfaces with prescribed mean curvature depending on the Gauss map.
- Surfaces in Euclidean space modeling capillarity problems: stability, geometry, Plateau-Rayleigh instability, bifurcation of liquid drops.
- Curves with special properties: slant helices, conformal trajectories.
- The hanging chain problem, including geometric aspects in architecture.
- The Euler problem of minimization of the moment of inertia: curves and extension on hypersurfaces..
- The Newton's problem of minimal resistance.