International geometric analysis conference in Milan

Titles and abstracts

Monday

Yannick Sire (Johns Hopkins University)

Parabolic gluing in geometric flows

I will present recent results and techniques to construct solutions to many parabolic geometric flows blowing up in finite or infinite time. This set of techniques is based on a new gluing construction, reminiscent to some extent to elliptic gluing in geometric analysis. Though the construction is quite technical, I will try to explain the general strategy and how to implement it so that it becomes clear that one can deal with many geometric PDEs enjoying bubbling behaviour. Several applications will be mentioned and I will also draw some possible future directions of research and open problems.

Luca Spolaor (UC San Diego)

Regularity Theory for Semicalibrated Integral Currents

In this talk I will survey old and new results about the regularity of semicalibrated integral currents, with particular focus on the recent proof of the rectifiability of their singular set. This is based on joint work with Minter, Parise and Skorobogatova.

Reinaldo Resende (Carnegie Mellon University)

On the boundary regularity of area minimizing currents

We will explore the state of the art in interior and boundary regularity for solutions to the oriented Plateau problem, specifically within the framework of integral currents. After reviewing recent developments in interior regularity, we will shift our focus to the boundary setting. There, we will discuss the types of boundary points that naturally arise in the theory and present some of the latest results on boundary regularity, highlighting open problems that remain unsolved. If time permits, I will outline the key ideas behind some of the proofs. This talk is based on joint work with Ian Fleschler.

Marco Badran (ETH Zurich)

Singular harmonic maps and nonlocal minimal surfaces in codimension two

The existence and regularity theory of minimal submanifolds has been a central topic in geometric analysis over the last sixty years. While the codimension one case is now well understood — thanks to extensive progress over the past decade — the codimension two case remains more elusive in many aspects. In this talk, I will discuss a novel nonlocal approximation scheme for higher codimension minimal surfaces, recently introduced by J. Serra, and explain a second-order self-interaction mechanism of the associated critical points. This phenomenon is intimately related to the structure and classification of singular (classical) harmonic maps into the circle.

Tuesday

Irene Fonseca (Carnegie Mellon University)

Phase Separation in Heterogeneous Media

Modern technologies and biological systems, such as temperature-responsive polymers and lipid rafts, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations using a variational approach based on the gradient theory of phase transitions, the potential and the wells may have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the critical case case when the scale of the small heterogeneities is of the same order of the scale governing the phase transition and the wells are fixed, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. The supercritical case for fixed wells is also addressed,. In the subcritical case and with moving wells, in which the phase separation scale is finer than the homogenization scale, the periodic homogenization enters into the energy through the double well potential, and the problem is proven to exhibit a separation of scales. Again, we show that the Gamma-limit is a truly anisotropic surface energy. Furthermore, the integrand of the surface energy is determined by a cell-problem taken over large cells, but it does not involve the concept of a plane-like minimizer. This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands) and Likhit Ganedi (Aachen University, Germany), USA).

Robin Neumayer (Carnegie Mellon University)

Quantitative Resolvent and Eigenfunction Estimates for the Faber-Krahn Inequality

For a bounded open set in Euclidean space with the same volume as the unit ball, the classical Faber-Krahn inequality says that the first Dirichlet eigenvalue of the Laplacian is at least that of the unit ball. We prove that the deficit in the Faber-Krahn inequality controls the square of the distance between the resolvent operator for the Dirichlet Laplacian of a given set and the resolvent operator on the nearest unit ball, as well as the squared L2 norm between kth eigenfunctions on this set and on the nearest ball for every k. This talk is based on joint work with Mark Allen and Dennis Kriventsov.

Paul Minter (Stanford University)

Stationary Integral Varifolds near Multiplicity 2 Planes

I will discuss work (joint with Spencer Becker-Kahn and Neshan Wickramasekera) which studies, in arbitrary dimension and codimension, the behaviour of stationary integral varifolds near multiplicity 2 planes. Assuming a certain topological structural condition holds on the (relatively open) set where the density of the varifold is strictly less than 2, we are able to show a version of Allard's regularity theorem holds near multiplicity 2 planes, with the structure being described by a Lipschitz 2-valued graph with improved estimates. Our proof also applies to varifolds with p-integrable generalised mean curvature for p strictly larger than the dimension of the varifold.

Filippo Gaia (ETH Zurich)

Singularities of Hamiltonian Stationary Lagrangian Surfaces

I will discuss the existence and regularity of critical points of the area functional among Lagrangian surfaces in symplectic 4-manifolds, reviewing classical results by R. Schoen and J. Wolfson, as well as recent progress by A. Pigati and T. Rivière. I will then present a variational construction of Hamiltonian stationary Lagrangian branched immersions in C^2 with multiple singularities at prescribed points, based on the sharp Wente inequality and an asymptotic analysis near the singularities. This is based on joint work with G. Orriols and T. Rivière. If time permits, I will also discuss a recent construction of Hamiltonian stationary branched immersions with infinitely many singularities but finite total area.

Raghavendra Venkatraman (University of Utah)

Spectral bounds rates of convergence of the graph laplacian on random geometric graphs

This talk has two parts. In the first, we study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution rho supported on a manifold M. The operators discussed in the paper are relevant in unsupervised learning and in particular are obtained by taking suitable scaling limits of widely used graph Laplacians over data clouds. We study the minimax risk for this eigenpair estimation problem and explore the rates of approximation that can be achieved by commonly used graph Laplacians built from random data. More concretely, assuming that rho belongs to a certain family of distributions with controlled second derivatives, and assuming that the d-dimensional manifold M where rho is supported has bounded geometry, we prove that the statistical minimax rate for approximating eigenvalues and eigenvectors in the H^1(M)-sense is n^(-2/(d+4)), a rate that matches the minimax rate for a closely related density estimation problem. To the best of our knowledge, our results are the first statistical lower bounds for this type of eigenpair estimation problem. In the second part, we then revisit the literature studying Laplacians over proximity graphs in the large data limit and prove that eigenpairs of these graph-based operators can induce manifold agnostic estimators with an error of approximation that, up to logarithmic corrections, matches our minimax lower bounds, providing in this way a concrete statistical basis for the claim that graph Laplacian based estimators are, essentially, optimal for this estimation problem. Time permitting, we indicate how these results draw inspiration from quantitative stochastic homogenization, and compare the rates of convergence in the case of sparse graphs (high contrast) and dense graphs (low contrast), and close with providing a flavor of the proofs of these results. Joint work with Nicolas Garcia Trillos and Chenghui Li (Wisconsin), and Scott Armstrong (Courant/Sorbonne).

Wednesday

Yangyang Li (University of Chicago)

Existence of 5 minimal tori in 3-spheres of positive Ricci curvature

In 1989, Brian White conjectured that every Riemannian 3-sphere contains at least five embedded minimal tori. The number five is optimal, corresponding to the Lyusternik-Schnirelmann category of the space of Clifford tori. I will present recent joint work with Adrian Chu, where we confirm this conjecture for 3-spheres of positive Ricci curvature. Our proof is based on min-max theory, with heuristics largely inspired by mean curvature flow.

Chao Li (Courant Institute, NYU)

Covering instability for the existence of positive scalar curvature metrics

I will present infinitely many closed non-orientable manifolds, such that each of them admits no Riemannian metric with positive scalar curvature, but its orientation double cover does. These examples were first studied to exhibit non-locality of Urysohn widths. The proof of the nonexistence result uses minimal hypersurfaces in non-orientable manifolds.

Guido De Philippis (Courant Institute, NYU)

Min-max construction of anisotropic minimal hypersurfaces

We use the min-max construction to find closed hypersurfaces which are stationary with respect to anisotropic elliptic integrands in any closed n-dimensional manifold. These surfaces are regular outside a closed set of zero n-3 dimension. The critical step is to obtain a uniform upper bound for density ratios in the anisotropic min-max construction. This confirms a conjecture posed by Allard. the talk is based on a joint work with A. De Rosa and Y. Li.

Thursday

Tristan Rivière (ETH Zurich)

Area Variations under Lagrangian and Legendrian constraints: Analytical Challenges and Geometric Motivations

We will present the problem of studying the regularity of critical points of the area among Lagrangian surfaces in a symplectic 4 manifold. We will give an optimal result on this question obtained in collaboration with Alessandro Pigati. Then, in the second part of the talk, we will describe a minmax problem on the area of Gauss maps of immersions of surfaces into the 3 sphere related to the Willmore conjecture. After having identified the absolute lower bound of the area among Gauss maps of closed surface immersions into the 3-sphere, we will describe the behavior of minimizing sequences and a new result on this topic obtained in collaboration with Gerard Orriols. Finally, in the last part of the talk we will address existence and regularity issues related to this minmax problem in connection with the first part of the talk.

Alessandro Carlotto (Università di Trento)

Non-persistence of strongly isolated singularities, and geometric applications

In this lecture, based on recent joint work with Yangyang Li (University of Chicago) and Zhihan Wang (Cornell University), I will present a generic regularity result for stationary integral n-varifolds with only strongly isolated singularities inside N-dimensional Riemannian manifolds, in absence of any restriction on the dimension (at least two) and codimension. As a special case, we prove that for any n greater than 1 and any compact (n+1)-dimensional manifold M the following holds: for a generic choice of the metric g all stationary integral n-varifolds in (M,g) will either be entirely smooth or have at least one singular point that is not strongly isolated. In other words, for a generic metric only ``more complicated'' singularities may possibly persist. This implies, for instance, a generic finiteness result for the class of all closed minimal hypersurfaces of area at most 4 pi^2 - epsilon (for any epsilon>0) in nearly round four-dimensional spheres: we can thus give precise answers, in the negative, to the well-known questions of persistence of the Clifford football and of Hsiang's hyperspheres in nearly round metrics. The aforementioned main regularity result is achieved as a consequence of the fine analysis of the Fredholm index of the Jacobi operator for such varifolds: we prove on the one hand an exact formula relating that number to the Morse indices of the conical links at the singular points, while on the other hand we show that the same number is non-negative for all such varifolds if the ambient metric is generic.

Daniel Stern (Cornell University)

Existence of extremal metrics for Laplace eigenvalues and minimal surfaces of prescribed topology

Classic results of Yang-Yau and Li-Yau showed that on every closed surface of unit area, the first nonzero eigenvalue of the Laplacian is bounded above by a constant depending only on the topology, launching the study of shape optimization problems for Laplace eigenvalues on surfaces. I'll discuss recent work with Karpukhin and Petrides confirming that every closed surface admits a metric maximizing the first eigenvalue of the Laplacian, and its connection with new methods (developed with Karpukhin, Kusner, and McGrath) for producing embedded minimal surfaces of prescribed topological type in S^3 and S^4.

Hui Yu (National University of Singapore)

Minimizing cones in the Alt-Caffarelli-Phillips problem

The Alt-Caffarelli-Phillips problem is a family of elliptic free boundary problems. This family includes the Bernoulli problem and the obstacle problem as special cases. Despite exciting developments in recent years, many important questions remain open, especially about singular cones. In this talk, we discuss two questions along this direction. The first question concerns the critical dimension for the appearance of singularities. The second question concerns density bounds for contact sets of singular cones. This talk is based on joint works with Ovidiu Savin at Columbia University.

Matilde Gianocca (ETH Zurich)

Harmonic Maps, Minimal Surfaces, and Their Spectra in S^3

In this talk, I will present five open questions concerning harmonic maps and minimal surfaces in S^3. In particular, I will focus on the existence of harmonic maps from arbitrary Riemann surfaces into S^3 under various additional assumptions—such as fullness, conformality, absence of branch points, or embeddedness. Most of these questions remain unresolved. The aim of the talk is to explore the connections between them, especially through the role of the energy index, and to discuss what types of maps might arise/not arise from natural min-max constructions in this setting.

Friday

Bozhidar Velichkov (Università di Pisa)

On the boundary branching set of the one-phase problem

The talk is dedicated to some recent result about the local structure of minimizing free boundaries for the one-phase Bernoulli problem. The focus is on the structure of the contact set C between the fixed and the free boundaries and on the boundary branching set (which is the boundary of C). Precisely, we will show that, when the fixed boundary is analytic, the following holds:

- In dimension 2 we will use conformal maps techniques to show that the boundary branching set is discrete.

- In dimension d>2 we will show that the boundary branching set has Hausdorff dimension at most d−2. The approach we use is based on the (almost-)monotonicity of a boundary Almgren-type frequency function, obtained via regularity estimates and a Calderon-Zygmund decomposition.

The results presented in this talk are contained in a series of papers in collaboration with Lorenzo Ferreri (Pisa) and Luca Spolaor (San Diego).

Nicholas Edelen (University of Notre Dame)

Improved regularity for minimizing capillary hypersurfaces

Capillary surfaces model the geometry of liquids meeting a container at an angle, and arise naturally as (constrained) minimizers of the Gauss free energy. We give improved estimates for the size of the singular set of minimizing capillary hypersurfaces: the singular set is always of codimension at least 4 in the surface, and this estimate improves if the capillary angle is close to $0$, $\pi/2$, or $\pi$. For capillary angles that are close to $0$ or $\pi$, our analysis is based on a rigorous connection between the capillary problem and the one-phase Bernoulli problem. This is joint work with Otis Chodosh and Chao Li.