International geometric analysis conference in Milan II

Titles and abstracts

Gioacchino Antonelli - University of Notre Dame
Area of Hölder planar curves and the coarea formula for Lipschitz maps on the Heisenberg group

Let \gamma=(x,y) be a smooth curve in the plane. Its signed area is defined by S(\gamma) = 1/2 \int_\gamma (x dy − y dx). When \gamma is a simple closed curve, this agrees, up to sign, with the area it encloses. By Young integration, S(\gamma) is well defined for every \alpha-Hölder curve with \alpha>1/2. In this talk, I will discuss a new sharp existence result for S(\gamma) at the critical exponent \alpha = 1/2, under an additional square-summability assumption. I will explain how ideas behind this result lead to a proof of the coarea formula for Lipschitz maps from H^1 to R^2, where H^1 denotes the first sub-Riemannian Heisenberg group. This gives a vector-valued coarea formula in the sub-Riemannian setting for merely Lipschitz maps, in a regime that was beyond the reach of previously available methods in geometric measure theory. This is joint work with Robert Young (NYU).

Riccardo Caniato - Caltech
Area rigidity for the regular representation of surface groups

Starting from the celebrated results of Eells and Sampson, a rich and flourishing literature has developed around equivariant harmonic maps from the universal cover of Riemann surfaces into negatively curved target spaces. In particular, such maps are known to be rigid, meaning that they are unique up to natural equivalence. This rigidity property fails when the target space has positive curvature, and comparatively little is known in this framework. In this talk, given a closed Riemann surface with strictly negative Euler characteristic, and a unitary representation of its fundamental group on a separable complex Hilbert space H which is weakly equivalent to the regular representation, we aim to discuss a lower bound on the Dirichlet energy of equivariant harmonic maps from the universal cover of the surface into the unit sphere S of H, and to give a complete classification of the cases in which the equality is achieved. As an interesting corollary, we obtain a lower bound on the area of equivariant minimal surfaces in S, and we determine all the representations for which there exists an equivariant, area-minimising minimal surface in S. The subject matter of this talk is a joint work with Antoine Song (Caltech) and Xingzhe Li (Cornell University).

Daniel Ketover - Rutgers University
Minimal surfaces of all genera in three spheres

I will describe a construction of minimal surfaces of all genera at least 2 in Riemannian three-spheres with positive Ricci curvature.  The argument uses variational methods applied to 2g+3 parameter sweepouts of genus g surfaces.  In the round S^3 we conjecturally reproduce the Lawson surfaces discovered in 1970.  

Sławomir Kolasiński - University of Warsaw
Quadratic flatness and Regularity for Codimension-One Varifolds with Bounded Anisotropic First Variation

Consider an n-dimensional integral varifold V in open subset U of 𝐑ⁿ⁺¹, whose mean curvature computed with respect to some non-euclidean norm ϕ is bounded and whose support has locally finite n-Hausdorff measure. Such varifolds were studied by Allard in his 1986 paper, where he proved an anisotropic version of his famous regularity theorem. In this setting, however, regularity holds only on a very specific set S of points satisfying a flatness condition. In a joint work with Mario Santilli, we proved that this flatness conditions indeed holds on almost all of the support of V. To this end we exploit the fact that spt‖V‖ is an (n,h)-set, provide a formula for its Minkowski content, and show that one can touch spt‖V‖ at almost all points with exactly two mutually tangent balls. As a consequence, we also derive perpendicularity and locality of the generalised mean ϕ-curvature vector of V.

Yang Li - Cambridge University
Large mass limit of G2 and Calabi Yau monopoles

I will discuss some recent progress on the Donaldson Segal programme, and in particular how calibrated cycles (coassociative submanifolds, special Lagrangians) arise from the large mass limit of G2 and Calabi Yau monopoles.

Jason Lotay - University of Oxford
Translators in Lagrangian mean curvature flow

Lagrangian mean curvature flow has received significant attention primarily through the Thomas-Yau and Joyce conjectures, which make predictions concerning long-time behaviour and singularity formation.  Key example of singularity models in Lagrangian mean curvature flow are given by translating solitons.  I will describe what is known about these translators, including recent developments.

Andrea  Marchese - University of Trento
Lipschitz solvability of prescribed Jacobian and divergence for singular measures

In recent joint work with Alberti and Bate, we showed that if a measure on Euclidean space is singular with respect to the Lebesgue measure, then operators such as the gradient, the divergence, and the Jacobian determinant fail to be closable from the space of Lipschitz functions to the corresponding L^p spaces, in the sense that limits of these operators along sequences of uniformly bounded Lipschitz functions are not determined by the limit of the functions themselves. In this talk I will explain how the same ideas can be extended to produce Lipschitz solutions with merely bounded data to the corresponding equations in a Lusin-type sense, seemingly in contrast with the philosophy behind Ornstein’s non-inequalities. I will also discuss connections with the structure of metric currents."

Massimiliano Morini - University of Parma
Elliptic regularisation of anisotropic mean curvature flows

Starting from Ilmanen’s pioneering work in the isotropic case, this talk discusses the extension of elliptic regularization to anisotropic mean curvature flow, a problem that has remained open because of serious technical difficulties specific to the anisotropic setting. After a brief introduction to the level-set formulation, we describe the anisotropic regularization scheme and prove, by an approach that is new even in the isotropic case, that under a non-fattening assumption the scheme converges in the Kuratowski sense to the corresponding non-fattening level-set evolution. This result, combined with a new conditional compactness theorem for anisotropic Brakke flows, is then used to show that every such generalized motion canonically supports a unique unit-density anisotropic Brakke flow. This is joint work  with C.A. Antonini, A. De Rosa, and S. Stuvard.

Robin Neumayer - Carnegie Mellon University
Rigidity of critical points of hydrophobic capillary functionals

The capillary energy functional models the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between 90 and 120 degrees must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between 90 and 180 degrees. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions. This talk is based on joint work with A. De Rosa and R. Resende.

Gerard Orriols - University of Cambridge
Large monopole Fueter sections and multivalued harmonic 1-forms

Recently there have been major advances in the program of Donaldson and Segal aiming to construct calibrated submanifolds as concentration sets of higher dimensional monopoles with diverging mass. Their full proposal speculates that a more precise relation should hold between gauge-theoretic enumerative invariants and appropriate counts of calibrated submanifolds, for which one should take into account the asymptotic behaviour of concentrated monopoles. This is governed by the Fueter equation, a nonlinear Dirac-type equation. In joint work with Yang Li, we show that when appropriately normalised, unbounded sequences of Fueter sections converge strongly in W^{1,2} to Q-valued harmonic 1-forms, generalizing the recent work of Esfahani–Li to higher charge.

Davide Parise - Imperial College London
Generic regularity of isoperimetric regions in dimension 8

Isoperimetric regions arise as minimisers of boundary area for a fixed enclosed volume, with sharp regularity theory, as established by Gonzalez, Massari, and Tamanini. In particular, the boundary of such a region is a smooth hypersurface away from a closed singular set of codimension seven. In closed Riemannan manifolds of dimension eight, this singular set consists of at most finitely many isolated points, with explicit singular examples having been constructed recently. In this talk, I will explain how under some assumptions on the choice of the ambient metric and enclosed volume, these singularities can be perturbed away, thus implying that the boundary is (generically) a smooth hypersurface. This is based on joint work with Kobe Marshall-Stevens and Gongping Niu. 

Tristan Rivière - ETH Zürich
Topological Singularities of  $S^2$ valued Harmonic Maps

In the early 80's Richard Schoen proposed to consider the Hilbert XXth problem for the Dirichlet Energy of maps taking values into $S^2$ or into a given closed sub-manifold of an euclidian space in general : does every homotopically trivial data at the boundary of a ball admits a smooth harmonic map extension ? We shall review the contributions made to this question in the last 45 years and illustrate the difficulty posed by the possible existence of topological singularities in the $S^2$ case while  implementing Minmax Procedures for the Dirichlet Energy among approximable maps. We will explain how Gauge Theory can be used to overcome some of the difficulties.

Melanie Rupflin - University of Oxford
A sharp quantitative stability result near infinitely concentrated minimisers

We discuss the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree 1 maps from closed surfaces of positive genus into the unit sphere. For this variational problem it is natural to view configurations which consist of a constant map from the given domain and an infinitely concentrated rotation as generalised minimisers and we discuss how a suitably defined gradient flow allows us to change the geometry and topology of the domain in a well controlled manner in order to obtain sharp quantitative estimates on all key features of almost minimising maps: the scale of concentration, the H^1-distance to the nearest bubble on the concentration region and the H^1-distance to the nearest constant away from the concentration point. This is joint work with Sebastian Woodward.

Mario Santilli - University of L'Aquila
Alexandrov soap bubble theorem for crystals

In this talk, I discuss some recent progress on the problem of bubbling phenomena for almost critical points of anisotropic surface energies. In particular, given a sequence of uniformly convex norms $ \phi_h $ on $ \R^{n+1} $ converging  to an arbitrary norm $ \phi $, the main result proves rigidity of $ L^1 $-accumulation points of sequences of sets (of finite perimeter) $ E_h \subseteq \R^{n+1} $, that are volume-constrained almost-critical points of the anisotropic surface energy associated with the norm $ \phi_h $.   Such limits are finite union of disjoint and possibly mutually tangent $ \phi $-Wulff shapes.

Lorenzo Sarnataro - University of Toronto
The Allen—Cahn equation and free boundary minimal surfaces

In recent years, the combined work of Guaraco, Hutchinson, Tonegawa, and Wickramasekera has established a min-max construction of minimal hypersurfaces in closed Riemannian manifolds, based on the analysis of singular limits of sequences of solutions of the Allen—Cahn equation, a semi-linear elliptic equation arising in the theory of phase transitions. In this talk, I will describe some recent boundary regularity results for such limit-interfaces, which provide the first steps towards an Allen—Cahn min-max construction of free boundary minimal hypersurfaces in Riemannian manifolds with boundary. This is based on joint works with Akashdeep Dey, Wenkui Du, Martin Li, and Davide Parise.

Felix Schulze - University of Warwick
PIC1 pinched manifolds are flat or compact

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, was resolved recently using Ricci flow. In this talk I will explain a direct analogue of that result in all dimensions. One aspect of the work is a new lifting technique that allows us to handle manifolds that are collapsed at infinity; until now this could only be achieved in 3D via work of Lott. The new theorem builds on earlier work of Deruelle, Simon and the speaker and separately of Lee-Topping. Joint work with Alix Deruelle, Man Chun Lee, Miles Simon and Peter Topping.

Anna Skorobogatova - ETH Zurich
Locally minimizing clusters: non-uniqueness and asymptotic behavior

Optimal bubble cluster problems concern the study of partitions of $\mathbb{R}^n$ into a finite collection of chambers, some with finite volume and some with infinite volume. One looks for local minimizers of interfacial area subject to volume constraints on the finite-volume chambers. The case of one infinite-volume chamber is the classical multiple bubble problem and has received much attention in recent decades, with a well-known conjecture of Sullivan predicting the existence of a unique minimizing configuration when there are not too many chambers, which has been partially verified to be true in low dimensions. We study a variant of the multiple bubble problem with more than one infinite-volume chamber, in particular the simplest case of 1 finite-volume chamber and 2 infinite-volume chambers. Here, Bronsard & Novack showed that uniqueness of local minimizers also holds in sufficiently low dimensions. In stark contrast, we show that uniqueness fails in a large number of dimensions starting from 8, and we provide a partial characterization of the asymptotic behavior at infinity that local minimizers can exhibit in these higher dimensions. This is based on joint work with Lia Bronsard, Robin Neumayer and Michael Novack.

Riccardo Tione - Università di Torino
Hyperbolic regularization effects for degenerate elliptic equations

In this talk, I will address a question raised by D. De Silva and O. Savin concerning the regularity of planar solutions to $div(Df(Du)) = 0$ under the sole assumption that $f \in C^1$ is strictly convex and $u$ is Lipschitz continuous. After a brief overview of the literature on such highly degenerate equations, starting from the foundational work of De Silva and Savin, I will focus on new regularity results obtained in collaboration with X. Lamy. These show, essentially, that $u$ is $C^1$ regular up to an isolated set of points provided $f$ is fully degenerate only along $C^1$ curves, an extension of the previously known results that required $f$ to be fully degenerate only at isolated points. I will explain some of the main ideas of the work, in particular the connection of this problem with Hamilton-Jacobi equations, and, if time allows, some details of the arguments.

Yoshihiro Tonegawa - Institute of Science Tokyo
Existence and regularity of multi-phase mean curvature flow in the hyperbolic space

We report some existence and regularity results on the multi-phase mean curvature flow in the standard hyperbolic space of general dimensions. Under a mild regularity assumption on the initial data, we prove the global-in-time existence and regularity results of the mean curvature flow. In particular, we show that the smoothness of the asymptotic boundary of the mean curvature flow persists for all time if the initial data is smooth. This reveals an interesting time-dependent regularity property different from the static case where the dimension plays an important role. This is a joint work with Qing Han and Nan Wu (University of Notre Dame).

Hui Yu - National University of Singapore
Generic regularity of the free boundary in the Alt-Caffarelli-Phillips problem

The Alt-Caffarelli problem and the Alt-Phillips problem are among the most well-studied elliptic free boundary problems. Much effort has been devoted to estimating the size of the singular set on the free boundary, for instance, in terms of its Hausdorff dimension. In this talk, we discuss how such estimates can be improved for generic boundary data. This talk is based on a recent joint work with Xavier Fernández-Real at EPFL.

Zihui Zhao - Johns Hopkins University
Singularity models for the one-phase free boundary problem

Free boundary problems are problems of solving an elliptic or parabolic PDE, but on a domain that is determined by the solution, and they arise naturally in the study of water waves and the propagation of flames, etc. In this talk, we will focus on the regularity of the one-phase free boundary problem, firstly studied by Alt and Caffarelli. In particular, I will highlight some of its connection with the regularity theory of minimal surfaces, including new examples of singularity models from isoparametric hypersurfaces, local and global asymptotics of solutions to the problem near points modelled on a homogeneous solution with isolated singularity.

Jonathan Zhu - University of Washington
Recent progress on capillary minimal surfaces

Capillary minimal surfaces have received a rise in attention from geometers and analysts as a bridge between minimal surface theory and free boundary problems in PDE. We’ll discuss recent progress on variational properties and Liouville/Bernstein type results for capillary minimal surfaces, particularly from a geometric perspective, as well as related work on strong intersection properties for general minimal surfaces.