Anisotropic geometric variational problems: existence, regularity and uniqueness

Project Information

ANGEVA

Grant agreement ID: 101076411

DOI

10.3030/101076411

EC signature date 22 November 2022

Start date 1 September 2023

End date 31 August 2028

Funded under

• European Research Council (ERC)

Total cost

€ 1 492 700,00

EU contribution

€ 1 492 700,00

Coordinated by

UNIVERSITA COMMERCIALE LUIGI BOCCONI
Italy

Periodic Reporting for period 1 - ANGEVA (Anisotropic geometric variational problems: existence, regularity and uniqueness)

Reporting period: 2023-09-01 to 2026-02-28

Summary of the context and overall objectives of the project

Minimal surfaces, the critical points of the area functional, have been central to geometry and analysis for more than a century. They arise in diverse contexts ranging from Plateau’s problem for soap films to deep conjectures in topology and general relativity. Yet the isotropic area functional is only an approximation: in many natural models, surface energy depends on direction or position. This anisotropy is fundamental in crystallography, capillarity, gravitational fields, and homogenization phenomena. Despite this wide relevance, the mathematical theory of anisotropic minimal surfaces remains at an early stage. Classical tools that underpin the isotropic case, such as monotonicity formulas, break down, leaving basic questions of existence, regularity, and uniqueness unresolved.

ANGEVA aims to build a comprehensive framework for anisotropic geometric variational problems. A first objective is to establish existence results for minimizers of anisotropic energy minimizers. A second direction is to develop the regularity theory for anisotropic minimal surfaces, and to study the fine structure of singularities. This analysis is also crucial for constructing anisotropic Brakke flows via elliptic regularization or Allen-Cahn approximation. A third major objective is to advance the min-max theory in closed manifolds: while the existence of isotropic minimal hypersurfaces is by now classical, the anisotropic counterpart has resisted progress since the 1980s. Finally, the project seeks to prove rigidity and stability results for critical points of anisotropic isoperimetric-type problems, with applications in volume-preserving anisotropic geometric flows.

By addressing these objectives, the project aims not only to close a fundamental gap in geometric analysis, but also to provide techniques of broad applicability, which are expected to yield new insights even in the isotropic theory. Beyond pure mathematics, anisotropic models underpin key phenomena in physics, materials science, and engineering. Establishing a rigorous analytical foundation for these problems will thus have an impact both in mathematics and in the modelling of real-world systems.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

During the first two years, the project has advanced on several of the core objectives outlined in the proposal.

One of the central novelties of this project is the extension of the min-max theory to the anisotropic setting. Together with De Philippis and Li, we showed existence and sharp regularity of closed anisotropic minimal hypersurfaces in closed manifolds of any dimension, a problem left open since the

1980s. This work extends Almgren-Pitts theory to the anisotropic setting and settles one of the major objectives of the project.

A second major direction concerned the regularity theory for anisotropic minimal surfaces. In collaboration with Lei and Young, we constructed polyhedral chains with distribution of tangent planes close to prescribed measures on the Grassmannian. In case the measure is supported on positively oriented planes, the polyhedral chains are Lipschitz multigraphs. We applied this construction to prove that, for an anisotropic integrand, polyconvexity is equivalent to quasiconvexity of the associated Q-integrands and to show that strict polyconvexity is necessary for the validity of a rectifiability theorem for anisotropic minimal surfaces.

A third line of research concerns the anisotropic isoperimetric problem. With Neumayer, I characterized small volume local minimizers of the anisotropic isoperimetric problem in closed manifolds, extending the classical Wulff framework. In parallel, with Tione, I characterized the stationary configurations for the double and triple bubble problem in the convex case, developing tools that can be adapted to more general anisotropic multi-phase problems.

A fourth area of activity has been anisotropic geometric flows and their approximations. My postdoc De Gennaro, together with coauthors, studied the asymptotic of Mullins-Sekerka and area-preserving curvature flows on the planar flat torus, and introduced variationally consistent discrete diffusion and redistancing numerical schemes for mean curvature flow, ensuring their stability and convergence.

Together, these advances represent substantial progress on ANGEVA’s four pillars, offering solutions to long-standing questions and tools for the next stages of the project.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

Our anisotropic min-max construction opens several research directions. First, we aim to strengthen the anisotropic min-max with a control on the topology, to obtain genus bounds. Second, we plan an alternative min-max construction via Allen-Cahn approximation. Third, we will pursue higher-index solutions via multi-parameter sweepouts, producing a richer landscape of critical points. Finally, we will investigate the existence of infinitely many anisotropic minimal hypersurfaces in closed manifolds, connecting to an anisotropic counterpart of Yau’s conjecture. Together, these directions will push the anisotropic min-max theory closer to the level of depth of its isotropic counterpart.

Our construction of Lipschitz multigraphs with almost prescribed tangent planes creates the question of whether all values of the multigraphs are needed. This would allow us to prove the existence of quasiconvex Q-integrands that are not polyconvex, a central question in the calculus of variations.

In the isoperimetric problem, our next goal is a rigidity theorem among finite perimeter sets of critical points of the capillary problem, both isotropic and anisotropic. Existing results rely heavily on smoothness assumptions, particularly at the capillary boundary. Our approach develops geometric measure theory tools to treat all finite-perimeter sets, without further structural or regularity assumption. As an application, we will analyze the asymptotic behavior of the volume-preserving capillary flow, aiming to prove convergence toward canonical equilibrium shapes.

Finally, we are developing a rigorous construction of anisotropic Brakke flows and approximation schemes. The goal is to extend Ilmanen’s elliptic regularization, originally for isotropic flows, to the anisotropic setting where monotonicity formulas fail. In parallel, we are building a framework based on parabolic anisotropic Allen-Cahn equation. This will allow singular evolutions to be treated within a rigorous variational framework. In the long term, anisotropic Brakke flows are expected to play the same foundational role in anisotropic geometric variational problems as in the isotropic case, enabling studies of stability, singularities, and asymptotic behavior. Beyond pure mathematics, such flows provide the natural language for describing the motion of crystal interfaces, the coarsening of grain boundaries, multiphase systems and other anisotropy-driven phenomena in materials science, physics and engineering.