Program and Abstracts

Costante Bellettini. Hypersurfaces with mean curvature prescribed by an ambient function.

Let N be a compact Riemannian manifold of dimension 3 or higher, and g a Lipschitz non-negative (or non-positive) function on N. In joint works with Neshan Wickramasekera, after developing a suitable compactness and regularity theory, we prove that there exists a two-sided (well-defined global unit normal) closed hypersurface M whose mean curvature attains the values prescribed by g. Except possibly for a small singular set (of codimension 7 or higher), the hypersurface M is C2 immersed; more precisely, the immersion is a quasi-embedding, namely the only nonembedded points are caused by tangential self-intersections (around such a non-embedded point, the local structure is given by two disks, lying on one side of each other, and intersecting tangentially, as in the case of two spherical caps touching at a point). The proof employs an Allen–Cahn approximation scheme, classical minmax, and gradient flow arguments. A key issue that appears in this construction, as well as in related compactness questions for PMC hypersurfaces, is the possible formation of ”hidden boundaries”: we will explore recent progress in these directions.

Romane Boutillier. Geometry of cable nets, minimal surfaces and architecture.

Minimal surfaces, which belong to the structural family of tensile surfaces, proved to have many advantages in architecture, especially for the construction of lightweight or temporary structures. They can be made of materials that have no bending stiffness but derive their resistance to normal pressure and forces from their curvature. This presentation summarizes the advantages of minimal surfaces in their architectural application.

Fran Burstall. Stability of harmonic maps to symmetric spaces.

I will describe some very old work on stable harmonic maps of surfaces into symmetric spaces. I exhume this material in part because it is pretty, in part because there have been recent developments and finally because there are still some unanswered questions.

Alessandro Carlotto. Free boundary minimal surfaces: advances and perspectives.

I will try to describe some of the things we have learnt, over the past few years, about three fundamental questions in the global theory of free boundary minimal surfaces. In essence, I will discuss how to construct such surfaces and how to distinguish them by virtue of the fine analysis of their (equivariant or absolute) Morse index. As a byproduct, we will sketch a comparative picture of variational vs. perturbative methods, and indicate what is still to be understood in that direction.

Jesús Castro Infantes. Genus one H-surfaces with k ends in H^2 x R

In this talk we will construct via a conjugate Jenkins-Serrin technique two families of properly Alexandrov-immersed constant mean curvature 0 < H ≤ 1/2 in H^2 x R with genus one and k ends asymptotic to vertical H-cylinders (for 0 < H < 1/2). Moreover, we will show the existence of properly Alexandrov-immersed H-surfaces with genus one and 2 ends displaying that there is not a Schoen-type theorem for immersed H-surfaces in H^2 x R. This is a joint work with J. Santiago.

Alberto Cerezo. Embedded free boundary CMC annuli in the unit ball.

Wente (1995) constructed the first known examples of non-rotational, free boundary annuli with constant mean curvature H≠0 in the unit ball B^3 ⊂ R^3. These annular surfaces were not embedded. 

With respect to the case H = 0, Fernandez, Hauswirth and Mira (2022) recently obtained examples of free boundary minimal annuli in B^3.

In this talk, we will discuss the construction of a one-parameter family of embedded CMC annuli with free boundary in B3 and a prismatic symmetry group of order 4n for any n > 1. They constitute the first known examples of embedded non-minimal CMC annuli with free boundary in the unit ball apart from Delaunay surfaces, and the first annular solutions to the partitioning problem in the ball that are not rotational. Additionally, other examples of CMC annuli that appear in this construction will be shown. This is a joint work with Isabel Fernandez and Pablo Mira.

Jean Chartier. Finding stationary nets on spheres using the disc flow method.

We construct stationary objects such as geodesics or geodesic nets on polyhedral or Riemannian spheres, using a discrete flow over sweepouts, known as the disc flow, in the manner of Joël Hass and Peter Scott.

Andrea Del Prete. The Jenkins-Serrin Theorem in Killing submersions and applications.

In 1966, Jenkins and Serrin found necessary and sufficient conditions on bounded domains of the plane that guarantee the existence and uniqueness of a minimal graph with possibly divergent values at the boundary. After giving the notion of Killing graphs in a 3-manifold with a Killing vector field, we describe how to approach the Jenkins-Serrin problem in this new setting using the divergence-lines technique. We also see some applications. This talk is based on a joint work with J. M. Manzano and B. Nelli.

Maria Fernanda Elbert. Hypersurfaces with constant higher order mean curvatures.

Let M^{n+1} be an oriented Riemannian manifold and φ : Σ^n → M an oriented hypersurface. The mean curvature of order k of the hypersurface, Hk, is defined as the kth symmetric function of the principal curvatures of φ : Σ^n → M. We recall that H = H1 is the mean curvature.

The stability of minimal hypersurfaces or CMC has drawn the attention of many mathematicians over the last few decades. Stability results for H1-hypersurfaces in the literature range from those applied to closed hypersurfaces to those applied to capillary hypersurfaces, which include free boundary hypersurfaces as a particular case.

In this talk, we will discuss the stability of Hk-hypersurfaces, exploring recent results with B. Nelli, R. de Lima and L. Damasceno.

José Espinar. Min-Oo conjecture for fully nonlinear conformally invariant equations.

In this talk we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard n sphere S^n under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we will work with.

This proves rigidity for compact connected locally conformally flat manifolds (M, g) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D(r), where D(r) denotes a geodesic ball of radius r ∈ (0, π/2] in S^n, and totally umbilical with mean curvature bounded bellow by the mean curvature of this geodesic sphere. Under the above conditions, (M, g) must be isometric to the closed geodesic ball D(r). In particular, we recover the solution by F.M. Spiegel to the Min-Oo conjecture for locally conformally flat manifolds.

Isabel Fernández. Free Boundary Minimal Annuli in the Ball.

We will present a family of free boundary minimal annuli immersed in the unit ball of Euclidean 3-space. These are the first such examples other than the critical catenoid, solving a problem posed by Nitsche in 1985. This is a joint work with Laurent Hauswirth and Pablo Mira.

Debora Impera. Rigidity and non-existence results for collapsed translators.

Translators for the Mean Curvature Flow give rise to a special class of eternal solutions, that besides having their own intrinsic interest, are models of a certain class of singularities that arise along the flow. Recently there has been a great effort in trying to construct examples of translators and in trying to classify them. In particular, many examples have been constructed of translators confined into slabs in space (collapsed translators), assuming that the slab width is greater than or equal to π. 

In this talk I will discuss some non-existence and rigidity results for collapsed translators when the slab width is sufficiently small. 

The talk is based on a joint project with N. M. Møller and M. Rimoldi.

Katrin Leschke. Periodic Darboux transforms.

In classical differential geometry, geometric transformations have been used to create new curves and surfaces from simple ones: the aim is to solve the underlying defining compatibility equations of curve or surface classes by finding solutions to a simpler system of differential equations arising from the transforms. Classically, the main concern was a local theory. In modern theory, global questions have led to a renewed interest in classical transformations. For example, in the case of a torus, the investigation of closing conditions for Darboux transforms naturally leads to the notion of the spectral curve of the torus. 

In this talk we discuss closing conditions for smooth and discrete polarised curves, isothermic surfaces and CMC surfaces. In particular, we obtain new explicit periodic discrete polarised curves, new discrete isothermic tori and new explicit smooth CMC cylinder.

Philipp Käse. A new family of CMC surfaces in homogeneous spaces.

In 1841 Delaunay characterized surfaces of constant mean curvature H = 1 in Euclidean 3-space invariant under rotation. The result was generalized by several authors to screw-motion invariant CMC surfaces in E(κ, τ ), but it turns out that the classification is not complete. In fact, new CMC surfaces arise in addition to the Delaunay family. In this talk I would like to present existence conditions and talk about further properties of these new surfaces.

Laurent Mazet. A rigidity result for free boundary minimal disks.

Thanks to the work of J. Jost, any convex Riemanian 3-ball contains either a minimal sphere or a free boundary minimal disk. If the Ricci curvature is non negative and the principal curvatures of the boundary are non less than 1, we will explain that it is then possible to control the perimeter of the free boundary minimal disk by 2π. Moreover the perimeter is equal to 2π if and only if the 3-ball is the Euclidean ball of radius 1. This is a joint work with Abraao Mendes.

Ilaria Mondello. Gromov-Hausdorff limits of manifolds with a Kato bound on the Ricci curvature.

The goal of this talk is to present some recent results on manifolds for which the negative part of the Ricci curvature satisfies a Kato bound, inspired by Kato potentials in the Euclidean space. This condition is implied for instance by a lower Ricci curvature bound, or an integral Ricci bound in the spirit of Gallot and Petersen-Wei. We will explain some analytic consequences of a Kato bound, and how we used them to study the structure of Gromov-Hausdorff limits of manifolds satisfying this kind of bounds. This talk is based on a joint work with G. Carron and D. Tewodrose.

David Moya. Results about Morse index in the Schwarzschild space.

We present a new family of properly embedded rotationally symmetric free boundary minimal surfaces in the Schwarzschild space which intersect the horizon at a height t0. This one parameter family of surfaces converges to Σ0 = {x ∈ R^n; xn = 0, |x| ≥ R0 ∈ R^+} when t0 → 0. In 2021 R. Montezuma proved that the Morse index of Σ0 in the Schwarzschild space of dimension 3 is one. We are going to extend this result to the generalized k-Schwarzschild space under certain assumptions. This is a joint work with Ezequiel Barbosa.

Frank Pacard. Doubling construction for embedded constant mean curvature surfaces in 3-manifolds.

I will describe some construction of embedded constant mean curvature surfaces in Riemannian 3-manifolds starting from an embedded minimal surface. This construction provides constant mean curvature surfaces whose genus is at least twice the genus of the minimal surfaces used for the construction.

Antonio Ros. The first eigenvalue of the Laplacian on closed surfaces.

The eigenvalues of the Laplacian of a compact surface are linked in different ways to the geometry of the surface. We are interested in estimates of the first eigenvalue that depend on the topology of the surface. The initial result was obtained by Hersch in the 1970s, who showed that for any metric on the sphere the product of the area and the first eigenvalue is less than or equal to 8π and the equality hold only for the constant curvature metric.

Over a compact surface of genus g > 0 there is a close relationship between the maximum of this functional, Λ1(g), and minimal immersions of the surface in a sphere S^n(1).

Among other results, we will present some recent properties on the asymptotic behaviour of the sequence {Λ1(g)}g.

Melanie Rothe. Lawson’s Bipolar Minimal Surfaces in the 5-Sphere.

Unlike Euclidean space R^n, the n-sphere S^n allows for closed minimal surfaces. Interestingly, the latter turned out to be highly relevant in the geometric optimization of Laplacian eigenvalues or the Willmore energy under fixed topology. However, regarding each topological class, the list of known examples is very limited. Concerning higher codimensions, H. B. Lawson has shown in 1970 that every minimal surface in S^3 induces a minimal surface in S^5, its so-called bipolar surface. For closed surfaces, the example of the bipolar Lawson surfaces τm,k ⊆ S^5 showed that the topology can thereby undergo a change. From this perspective, we give a topological classification of the bipolar Lawson surfaces ξm,k and ηm,k. Additionally, we provide upper and lower area bounds, and find that these surfaces are not embedded for m ≥ 2 or k ≥ 2.

Marcos Paulo Tassi. Analytic saddle spheres in S^3 are equatorial.

The Calabi-Almgren Theorem states that any minimal topological sphere immersed in the 3-sphere S^3 must be equatorial, i.e., a totally geodesic 2-sphere of S^3. In this talk we present a generalization of the Calabi-Almgren Theorem: any real analytic topological sphere immersed in S^3 which is also saddle (i.e., its principal curvatures k1, k2 satisfy k1k2 ≤ 0 at each point of the surface) must be equatorial. We also present a sketch of its proof, based on the study of the umbilical points of the saddle sphere and the Poincaré-Hopf Index Theorem and we discuss smooth, non-equatorial counter-examples. 

This is a joint work with J.A. Gálvez and P. Mira.

Francesca Tripaldi. Sequences of properly embedded minimal surfaces.

This talk centres around the study of sequences of minimal surfaces in a three manifold. So far, an in-depth characterisation of such sequences has been carried out in the case where the surfaces considered are discs, both in regards to the curvature blow-up points and the limit surfaces that can arise. In order to study the topology of the limit leaves of sequences of properly embedded minimal discs, Bernstein and Tinaglia first introduced the concept of the simple lift property, since such limit leaves satisfy this property. They proved that an embedded surface Σ ⊂ Ω with the simple lift property must have genus zero, if Ω is an orientable three-manifold satisfying certain geometric conditions. During my PhD, I generalised this result by considering an arbitrary orientable three-manifold Ω and proving that one is still able to restrict the topology of an arbitrary surface Σ ⊂ Ω with the simple lift property. In this talk, I will present which topological results can be extended in the case of sequences of minimal surfaces of arbitrary finite genus.