Geometric Analysis is a field of Pure Mathematics that lies in the interplay between Geometry and PDEs and has applications to several branches of Mathematics such as Riemannian Geometry, Topology, and Complex Variables, as well to General Relativity, Crystallography, Material Science, Architecture, and other fields. Among many other achievements, the solutions of the Positive Mass Conjecture in General Relativity, the Poincaré Conjecture in Topology, the Lawson, Willmore and Yau Conjectures in Differential Geometry, or the Kobayashi Conjecture in Complex Geometry show the current relevance and usefulness of Geometric Analysis in Pure Mathematics. Numerous well-established groups work in this field in several countries, which makes this topic highly-competitive.
The project Geometric Analysis is supported by the
Ministerio de Ciencia, Innovación y Universidades via Agencia Estatal de Investigación
(PID2023-150727NB-I00, September 1, 2024 — August 31, 2028) and their principal investigators are
Joaquín Pérez and
Antonio Alarcón at the University of Granada. Our research team is composed by 8 researchers in different areas of Geometric Analysis working in different centers in Spain. As for the work team, we count on two Ph.D. students, a young researcher, and two senior researchers from abroad.
The broad goal of this project is to delve into the understanding of several problems in geometric analysis and its applications. We have extensive previous experience in some of them, backed by many papers published in top-level journals. Some others are fairly new and will lead us to enlarge our research lines. We shall focus on four subjects that are linked by strong synergies, namely:
A) Minimal surfaces.
A.1) Minimal surfaces from a geometric viewpoint.
A.2) Minimal surfaces and architecture.
A.3) Minimal surfaces from a complex analytic viewpoint.
B) Constant mean curvature surfaces.
B.1) CMC-1 surfaces in hyperbolic space: Bryant surfaces.
B.2) CMC surfaces in E(κ,τ)-spaces.
B.3) CMC surfaces in metric Lie groups.
B.4) CMC surfaces in Riemannian three-manifolds.
B.5) Stability of liquid channels.
C) Complex geometry.
D) PDE related problems.
D.1) Overdetermined elliptic problems.
D.2) The first Laplacian eigenvalue on compact surfaces.
This research project Geometric Analysis continues a series of projects, with the same title, which have been successively supported by different calls of the Spanish research program. The composition of our team has been slightly changing in each edition of the project. The principal investigator (PI) of the first research project Geometric Analysis was Antonio Ros (MTM2007-61775 from 2007 to 2012). The next PI was Joaquín Pérez (MTM2011-22547 from 2012 to 2014 and MTM2014-52368-P from 2015 to 2018). Antonio Alarcón was incorporated as second PI of the project for its latest editions (MTM2017-89677-P from 2018 to 2021, and PID2020-117868GB-I00 from 2012 to 2024).