The Schwarz Lemma, a cornerstone in complex analysis, offers crucial insights into the behavior of holomorphic functions on the unit disk. Beyond its classical formulation, this result has significant geometric implications, particularly in the context of minimal surfaces, which are surfaces with zero mean curvature. These surfaces arise in both differential geometry and physics, with applications ranging from soap films to general relativity. The conformality of harmonic maps, especially those that preserve area, is fundamental in understanding minimal surfaces, and the Schwarz Lemma provides key constraints on these maps. It helps in establishing curvature bounds and offers tools for analyzing the behavior of minimal surfaces near their boundaries. In this presentation, we will explore two key results that extend the reach of the Schwarz Lemma in the theory of minimal surfaces: (1) Extension of the Schwarz Lemma for Conformal Parameterization of Minimal Surfaces: We will introduce a generalized version of the Schwarz Lemma that applies to conformal mappings associated with minimal surfaces, offering a new perspective on their parameterization. (2) Solution to the Gaussian Curvature Conjecture for Minimal Graphs: We will present a solution to the long-standing conjecture related to the Gaussian curvature of minimal graphs, shedding light on their intrinsic geometry. These results showcase the deep interplay between complex analysis and geometric properties of minimal surfaces, revealing new avenues for research and applications.

The Schwarz Lemma, a cornerstone in complex analysis, offers crucial insights into the behavior of holomorphic functions on the unit disk. Beyond its classical formulation, this result has significant geometric implications, particularly in the context of minimal surfaces, which are surfaces with zero mean curvature. These surfaces arise in both differential geometry and physics, with applications ranging from soap films to general relativity. The conformality of harmonic maps, especially those that preserve area, is fundamental in understanding minimal surfaces, and the Schwarz Lemma provides key constraints on these maps. It helps in establishing curvature bounds and offers tools for analyzing the behavior of minimal surfaces near their boundaries. In this presentation, we will explore two key results that extend the reach of the Schwarz Lemma in the theory of minimal surfaces: (1) Extension of the Schwarz Lemma for Conformal Parameterization of Minimal Surfaces: We will introduce a generalized version of the Schwarz Lemma that applies to conformal mappings associated with minimal surfaces, offering a new perspective on their parameterization. (2) Solution to the Gaussian Curvature Conjecture for Minimal Graphs: We will present a solution to the long-standing conjecture related to the Gaussian curvature of minimal graphs, shedding light on their intrinsic geometry. These results showcase the deep interplay between complex analysis and geometric properties of minimal surfaces, revealing new avenues for research and applications.

In this lecture, we address Serrin-type problems in Riemannian manifolds. We begin by establishing a Heintze-Karcher inequality and proving a Soap Bubble theorem, with its corresponding rigidity, in the context of ambient spaces with a Ricci tensor bounded from below. We then focus on Serrin’s problem within bounded domains of manifolds endowed with a conformal vector field. A key tool in this analysis is a new Pohozaev identity that incorporates the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. This lecture is based on joint works with A. Roncoroni (Politecnico di Milano, Italy), M. Santos (UFPB, Brazil), M. Andrade (UFS, Brazil), and Diego Marín (Universidad de Granada, Spain)

In this talk, we discuss the concept of stable extremal domains for the first Dirichlet eigenvalue of the Laplacian operator. We classify the stable extremal domains in the 2-sphere and higher-dimensional spheres when the boundary is minimal. Additionally, we establish topological bounds for stable domains in general compact Riemannian surfaces, assuming either nonnegative total Gaussian curvature or small volume. This is a joint work with Marcos P. Cavalcante (UFAL, Brazil).

We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Ampère equation for the conformal factor coupled with ODEs. We exhibit explicit solutions and obtain abstract existence results for complete expanding solitons and singular shrinking and steady ones. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics.

The Gauss map of a minimal surface in $\mathbb R^3$, parametrised as a conformal minimal immersion from an open Riemann surface $M$ into $\mathbb R^3$, is a meromorphic function on $M$. Although the Gauss map has been a central object of interest in the theory of minimal surfaces since the mid-19th century, it was only recently proved by Alarcón, Forstnerič and López, using new complex-analytic methods, that every meromorphic function on $M$ is a Gauss map. It remains an open problem to usefully characterise those meromorphic functions that are the Gauss map of a complete minimal surface. I will describe recent joint work with Antonio Alarcón, in which we take a new approach to this problem. We investigate the space of meromorphic functions on $M$ that are the Gauss map of a complete minimal surface from a homotopy-theoretic viewpoint, using a new h-principle as a key tool. My talk will include a brief general introduction to h-principles and their applications.

Is it always possible to park a car in a parking space whose size is exactly the same as the car? Can we do the same with a multi-trailer truck? In this talk, we will show the relationship between these two questions and the theory of distributions on differentiable manifolds. We will review this theory and focus our attention on bracket-generating distributions. Typical examples of this class of distributions are Contact and Engel structures. We will motivate these objects by showing other examples and establish several results about their tangent curves. In particular, we will show that the spaces of regular tangent knots are flexible if the dimension of the manifold is greater than $3$. These results are part of a joint work with Álvaro del Pino (Utrecht University).

What is the smallest volume a convex body $K$ in $\mathbb R^n$ can have for a given surface area? This question is in the reverse direction to the classical isoperimetric problem and, as such, has an obvious answer: the infimum of possible volumes is zero. One way to make this question highly non-trivial is to assume that $K$ is uniformly convex in the following sense. We say that $K$ is $\lambda$-convex if the principal curvatures at every point of its boundary are bounded below by a given constant $\lambda>0$ (considered in the barrier sense if the boundary is not smooth). By compactness, any smooth strictly convex body in $\mathbb R^n$ is $\lambda$-convex for some $\lambda>0$. Another example of a $\lambda$-convex body is a finite intersection of balls of radius $1/\lambda$ (sometimes referred to as ball-polyhedra). Until recently, the reverse isoperimetric problem for $\lambda$-convex bodies was solved only in dimension 2. In a recent joint work with Kateryna Tatarko, we resolved the problem also in $\mathbb R^3$. We showed that the lens, i.e., the intersection of two balls of radius $1/\lambda$, has the smallest volume among all $\lambda$-convex bodies of a given surface area. For $n>3$, the question is still widely open. I will outline the proof of our result and put it in a more general context of reversing classical inequalities under curvature constraints in various ambient spaces.

We classify those rotationally invariant translators of the mean curvature flow in the Hyperbolic Einstein Space-time \(\mathbb{H}^n\times_{-1}\mathbb{R}\). Next, we consider a connected, compact space-like translator whose boundary is the boundary of a bounded open domain in a slice. If the domain is invariant by an isometry \(\sigma\) of \(\mathbb{H}^n\), then the traslator is invariant by \(\sigma\times id\). We then characterize one of the rotationally invariant examples.

In this talk we will introduce Gromov's $h$-principle theory from a basic and accessible perspective. We will motivate it through visual examples with special emphasis on the method of Convex Integration. Many problems in Differential Topology involve differential relations (differential equations, inequalities, etc.). In many contexts, it can be proven that there exists an $h$-principle: this means that the resolution of certain geometric problems can be reduced to studying the underlying Algebraic Topology. We will show how these techniques can be applied to the study of maximal growth distributions on smooth manifolds. Prototypical examples of these objects are Contact and Engel structures.

In this talk, our goal is to discuss the existence of at least two distinct conformal metrics with prescribed gaussian curvature and geodesic curvature respectively, $K_{g}= f + \lambda$ and $k_{g}= h + \mu$, where f and h are nonpositive functions and \lambda and \mu are positive constants. Utilizing Struwe's monotonicity trick, we investigate the blowup behavior of the solutions and establish a non-existence result for the limiting PDE, eliminating one of the potential blow-up profiles.

The Allen–Cahn equation is a semilinear parabolic partial differential equation that models phase-separation phenomena and which provides a regularization for the mean curvature flow (MCF), one of the most studied geometric flows. In this talk, we employ Morse-theoretical considerations to construct eternal solutions of the Allen–Cahn equation that connect unstable equilibria in compact manifolds. We describe the space of such solutions in a round 3-sphere under a low-energy assumption, and indicate how these solutions can be used to produce geometrically interesting eternal MCFs. This is joint work with Jingwen Chen (University of Pennsylvania).

A compact flat Lorentzian manifold is the quotient of the Minkowski space by a discrete subgroup \(\Gamma\) of the isometry group, acting properly, freely and cocompactly on it. A classical result by Goldman, Fried and Kamishima states that, up to finite index, \(\Gamma\) is a uniform lattice in some connected Lie subgroup of the isometry group, acting properly and cocompactly, generalizing Bieberbach theorem to the Lorentzian signature. Such compact quotients are called "standard". More generally, a compact quotient of a homogeneous space \(G/H\) of a Lie group \(G\) is standard if the fundamental group action extends to a proper cocompact action of a connected Lie subgroup of \(G\). It turns out that looking for standard quotients is an easier problem when studying the existence of compact quotients of homogeneous spaces. This talk is about compact locally homogeneous plane waves. Plane waves can be thought of as a deformation of Minkowski spacetime, they are of great mathematical and physical interests. In this talk, we describe the isometry group of a 1-connected homogeneous non-flat plane wave, and show that compact quotients are “essentially" standard. As an application, we obtain that the parallel flow of a compact plane wave is equicontinuous. This is a joint work with M. Hanounah, I. Kath and A. Zeghib.

Translators in \(\mathbb{R}^3\) are solitons of the mean curvature flow for embedded 2-surfaces. In the semigraphical case, where the translators are allowed graphical as well as vertical components, Hoffman-Martín-White classified the surfaces into six types. They conjectured the uniqueness of the objects within two families contained in slabs, the "helicoids" and the "pitchforks," for any given width. We present the proof of the conjecture by combining an arc-counting argument motivated by Morse-Radó theory for translators with a rotational application of the maximum principle. We then discuss applications of this result to the classification of semigraphical translators in \(\mathbb{R}^3\) and their limits, related to the work of Hoffman-Martín-White and Gama-Martín-Møller. This is joint work with F. Martín and M. Sáez.