Geometric Variational Problems (PID2023-151060NB-I00) is a 4-year project (January 1, 2025 – December 31, 2028) at the University of Granada funded by MICIU-AEI and FEDER-UE. The principal investigators are Manuel Ritoré and César Rosales (University of Granada). The research team is also formed by Antonio Cañete (University of Sevilla) and Ana Hurtado (University of Granada). The work team is formed by Gianmarco Giovannardi (Università di Firenze), Andrea Pinamonti (Università di Trento) and Julián Pozuelo (Università di Padova).

The current project is the continuation of several previous projects where Manuel Ritoré was the principal investigator. Our research group began its journey in 2004 and has been working during the last 20 years as an individual project and sometimes in coordination with the project Geometric Analysis led by Vicente Palmer (and recently by Vicent Gimeno) at University Jaume I of Castellón. We form a small but dynamic group, which has evolved along time by incorporating some new members and research lines. The project is also part of the Spanish Network of Geometric Analysis.

For the most time, we have dealt with the optimization of geometric functionals in different metric measure spaces such as Riemannian and sub-Riemannian manifolds, or weighted manifolds. The corresponding study provides a fruitful interaction between different areas such as Calculus of Variations, Geometric Measure Theory and Partial Differential Equations. Besides its mathematical interest, the analysis of these problems has also applications in disciplines like Control Theory, Vision Theory, Crystallography, Materials Science and Geometric Optics.

The current research lines of the project are the following:

• Variational problems related to the sub-Riemannian and sub-Finsler area functionals.
• Isoperimetric problems in weighted manifolds and conductive Laplacians.
• Geometric inequalities for anisotropic functionals.