ERC StG ANGEVA

Anisotropic geometric variational problems: existence, regularity and uniqueness

ERC Starting Grant 2022 - Period: 2023 - 2028

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Project ID

Title: Anisotropic geometric variational problems: existence, regularity and uniqueness

Acronym: ANGEVA

Grant agreement ID: 101076411

DOI: 10.3030/101076411

Start date: 1 September 2023

End date: 31 August 2028

Duration: 60 months

Primary coordinator: Antonio De Rosa

Host institution: Bocconi University

Funding Agency

Funded under: European Research Council (ERC)

Funding Scheme: Starting Grant

Call year: 2022 Panel: PE1 Project number: 101076411

Total cost: € 1 492 700,00

Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Here you can find a divulgation article and here you can find a divulgation video about the project

Description

The focus of this project is to advance the theory of anisotropic geometric variational problems. A vast literature is devoted to the study of critical points of the area functional, referred to as minimal surfaces. However, minimizing the surface area is often an idealization in physics. In order to account for preferred inhomogeneous and directionally dependent configurations and to capture microstructures, more general anisotropic energies are often utilized in several important models. Relevant examples include crystal structures, capillarity problems, gravitational fields and homogenization problems. Motivated by these applications, anisotropic energies have attracted an increasing interest in the geometric analysis community. Moreover in differential geometry they lead to the study of Finsler manifolds. Unlike the rich theory for the area functional, very little is understood in the anisotropic setting, as many of the essential techniques do not remain valid. This project aims to develop the tools to prove existence, regularity and uniqueness properties of the critical points of anisotropic functionals, referred to as anisotropic minimal surfaces. In order to show their existence in general Riemannian manifolds, it will be crucial to generalize the min-max theory. This theory plays a crucial role in proving a number of conjectures in geometry and topology. In order to determine the regularity of anisotropic minimal surfaces, I will study the associated geometric nonlinear elliptic partial differential equations (PDEs). Finally, in addition to the stationary configurations, this research will shed light on geometric flows, through the analysis of the related parabolic PDEs. The new methods developed in this project will provide new insights and results even for the isotropic theory: in the size minimization problem, in the vectorial Allen-Cahn approximation of the general codimension Brakke flow, and in the Almgren-Pitts min-max construction.

Team

Postdocs:

Daniele De Gennaro (2024-2026)
Aria Halavati (2025-2027)
Ling Wang (2025-2027)

Open Positions

Two 24 months (renewable) postdoctoral research position at Bocconi University, Milan. Closed on: 30th November 2024
The results of the competition can be found here.
One 24 months (renewable) postdoctoral research position at Bocconi University, Milan. Closed on: 31st May 2024
The results of the competition can be found here.

Conferences

International geometric analysis conference in Milan II. June 8-12, 2026. Bocconi University, Milan, Italy. Co-organized with A. Pigati.
International geometric analysis conference in Milan. June 23-27, 2025. Bocconi University, Milan, Italy. Co-organized with A. Pigati.
Geometric Analysis & PDEs Christmas Workshop. December 18, 2024. Bocconi University, Milan, Italy. Co-organized with A. Pigati.
Geometric Measure Theory and applications 2024. June 17-21, 2024. Palazzone SNS, Cortona, Italy. Co-organized with G. Caldini, L. De Masi, A. Marchese and A. Massaccesi.

Publications

Rigidity of critical points of hydrophobic capillary functionals. A. De Rosa, R. Neumayer, and R. Resende. Submitted. 2025. ArXiv:2509.22532
Construction of fillings with prescribed Gaussian image and applications. A. De Rosa, Y. Lei, and R. Young. Archive for Rational Mechanics and Analysis. Volume 249, article number 39, 2025.
The double and triple bubble problem for stationary varifolds: the convex case. A. De Rosa, and R. Tione. Transactions of the American Mathematical Society. Volume 378, Number 5, Pages 3393–3444, 2025.
The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus. V. Arya, D. De Gennaro, and A. Kubin. Submitted. 2025. ArXiv:2503.05399
Elementary discrete diffusion/redistancing schemes for the mean curvature flow. A. Chambolle, D. De Gennaro, and M. Morini. Submitted. 2025. ArXiv:2506.05946
Variational nonlinear and nonlocal curvature flows. D. De Gennaro. Submitted. 2025. ArXiv:2506.05951
Direction of Impact for Explainable Risk Assessment Modeling. E. Borgonovo, M. Baucells, A. De Rosa, E. Plischke, J. Barr, H. Rabitz. Risk Analysis. Volume 45, Pages 2157–2182, 2025.
Existence and regularity of min-max anisotropic minimal hypersurfaces. G. De Philippis, A. De Rosa, and Y. Li. Submitted. 2024. ArXiv:2409.15232
Local Minimizers of the Anisotropic Isoperimetric Problem on Closed Manifolds. A. De Rosa, and R. Neumayer. Indiana University Mathematics Journal. 2024. ArXiv:2308.04565
On the Theory of Anisotropic Minimal Surfaces. A. De Rosa. Notices of the American Mathematical Society. Volume 71, Number 7, 2024 .