# Geometria em Lisboa Seminar

## Past sessions

### Sharp inequalities relating volume and the min-max widths of Riemannian three-spheres.

Min-max theories for the area functional have undergone ground-break developments in recent years. One aspect of these theories is that they define many notions of "width" that can be understood as geometric invariants of compact Riemannian manifolds. As such, it is an interesting question to understand what sort of geometric information they encode. In this talk, we will focus on the Simon-Smith width of Riemannian three-dimensional spheres, discussing how large it can be among metrics normalised to have the same volume, and necessary conditions for maxima under further restrictions. This is a joint work with Rafael Montezuma (UM-Amherst).

### Lifting Lagrangians from symplectic divisors

We prove that there are infinitely many non-symplectomorphic monotone Lagrangian tori in complex projective spaces, quadrics and cubics of complex dimension at least $3$. This is a consequence of the following: if $Y$ is a codimension $2$ symplectic submanifold of a closed symplectic manifold $X$, then we can explicitly relate the superpotential of a monotone Lagrangian $L$ in $Y$ with the superpotential of a monotone Lagrangian lift of $L$ in $X$. This sometimes involves relative Gromov-Witten invariants of the pair $(X,Y)$. We will define the superpotential, which is a count of pseudoholomorphic disks with boundary on a Lagrangian, and which plays an important role in Floer theory and mirror symmetry. This is joint work with D. Tonkonog, R. Vianna and W. Wu.

### Construction of harmonic mappings

Geometric variational problems frequently lead to analytically extremely hard, non-linear partial differential equations, where the standard methods fail. Thus finding non-trivial solutions is challenging. The idea is to study solutions with a certain minimum level of symmetry (i.e. group actions with low cohomogeneity), and use the symmetry to reduce the original problem to systems of non-linear ordinary differential equations, typically with singular boundary values. In my talk I explain how to construct harmonic mappings between manifolds with a lot of symmetry (i.e. cohomogeneity one manifolds). If time permits, I will discuss applications of the developed methods.

### On the shape of a Riemannian manifold with large first nonzero eigenvalue for the Laplacian and the Dirichlet-to-Neumann operator

Classically, there is a strong relationship between the shape of a Riemannian manifold and the spectrum of its Laplace-operator, and, more generally, with the spectrum of Laplace type operators. In particular, for the spectrum of the Laplacian, the presence of a large first nonzero eigenvalue is related to some concentration phenomena. In the first part of the talk, I will recall these classical relations. In the second part of the talk, I will introduce the Dirichlet-to-Neumann operator on a manifold with boundary. I will survey the same types of questions in this context (existence of large first nonzero eigenvalue, concentration phenomena) without going into the technical details. This second part corresponds to work in progress with Alexandre Girouard.

### Projectively induced Ricci-flat Kaehler metrics

The aim of this talk is to discuss the problem of classifying Kaehler-Einstein manifolds which admit an isometric and holomorphic immersion into the complex projective space. We start giving an overview of the problem focusing in particular on the Ricci-flat case. Ricci-flat non-flat Kaehler manifolds are conjectured to be not projectively induced. Next, we give evidence to this conjecture for Calabi’s Ricci-flat metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds.

### Introduction to Mirror Symmetry on the Hitchin System

This will be an introductory talk for the Working Seminar on Mirror Symmetry on the Hitchin System. During this minicourse, organized by T. Sutherland and myself, we aim to understand Mirror Symmetry on Higgs moduli spaces as a classical limit of the Geometric Langlands program. In this talk I will briefly describe the geometrical objects involved in this program and provide a motivation for it coming from mathematical physics. The structure of the working seminar will also be discussed.

### On a conjecture about curve semistable Higgs bundles

We say that a Higgs bundle $E$ over a projective variety $X$ is curve semistable if for every morphism $f : C \to X$, where $C$ is a smooth irreducible projective curve, the pullback $f^\ast E$ is semistable. We study this class of Higgs bundles, reviewing the status of a conjecture about their Chern classes.

### Stability of symplectic leaves

In this talk I will give a gentle introduction to Poisson manifolds, which can be thought of as (singular) symplectic foliations. As an illustration of the kind of problems one deals in Poisson geometry, I will discuss and give some results on stability of symplectic leaves.

### Constructions of contact manifolds with controlled Reeb dynamics

The Reeb flow of a contact form is a generalisation of Hamiltonian flows on energy hypersurfaces in classical mechanics. In this talk I shall address the question of how "complicated" such flows can be. Among other things, I plan to discuss a construction of Reeb flows with a global surface of section on which the Poincaré return map is a pseudorotation. This is joint work with Peter Albers and Kai Zehmisch

### Geography of (bi)linearized Legendrian contact homology

The study of Legendrian submanifolds in contact geometry presents some similarities with knot theory. In particular, invariants are needed to distinguish Legendrian isotopy classes. Linearized Legendrian contact homology is one of these, and is based on the count of holomorphic curves. It is obtained after linearizing a differential graded algebra using an augmentation. A bilinearized version using two augmentations was introduced with Chantraine.

After a self-contained introduction to this context, the geography of these invariants will be described. In the linearized case, it was obtained with Sabloff and Traynor. The bilinearized case turned out to be far more general and was studied with Galant.

### Equivariant and nonequivariant contact homology

I will explain how to make use of geometric methods to obtain three related flavors of contact homology, a Floer theoretic contact invariant. In particular, I will discuss joint work with Hutchings which constructs nonequivariant and a family floer equivariant version of contact homology. Both theories are generated by two copies of each Reeb orbit over $\mathbb{Z}$ and capture interesting torsion information. I will then explain how one can recover the original cylindrical theory proposed by Eliashberg-Givental-Hofer via our constructions.

### Spectrum of the Robin Laplacian: recent results, and open problems

I will discuss two recent theorems and one recent conjecture about maximizing or minimizing the first three eigenvalues of the Robin Laplacian of a simply connected planar domain. Conformal mappings and winding numbers play a key role in the geometric constructions. In physical terms, these eigenvalues represent decay rates for heat flow assuming a “partially insulating” boundary.

Joint with the CEAFEL seminar

### Hamiltonian $S^1$-spaces with large equivariant pseudo-index.

Let $(M,\omega)$ be a compact symplectic manifold of dimension $2n$ endowed with a Hamiltonian circle action with only isolated fixed points. Whenever $M$ admits a toric $1$-skeleton $\mathcal{S}$, which is a special collection of embedded $2$-spheres in $M$, we define the notion of equivariant pseudo-index of $\mathcal{S}$: this is the minimum of the evaluation of the first Chern class $c_1$ on the spheres of $\mathcal{S}$.

In this talk we will discuss upper bounds for the equivariant pseudo-index. In particular, when the even Betti numbers of $M$ are unimodal, we prove that it is at most $n+1$ . Moreover, when it is exactly $n+1$, $M$ must be homotopically equivalent to $\mathbb{C}P^n$.

### Symplectic triangle inequality

This talk will be concerned with handling problems about embedding Lagrangians in symplectic four-manifolds where the target manifold is rational. In particular, we will determine those three-fold blow-ups of the symplectic ball which admit an embedded Lagrangian projective plane.

### On Einstein spaces

A Riemannian manifold $(M,g)$ of dimension $n \ge 3$ is called an Einstein space if $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, where trivially $\lambda = \kappa$, with $\kappa$ the (normalized) scalar curvature; in this case one easily proves that $\lambda = \kappa = \operatorname{constant}$.

We recall the fact that any $2$-dimensional Riemannian manifold satisfies the relation $\operatorname{Ric} = \lambda \cdot \operatorname{id}$, but the function $\lambda= \kappa$ is not necessarily a constant. It is well known that any $3$-dimensional Einstein space has constant sectional curvature. Thus the interest in Einstein spaces starts with dimension $n=4$.

Singer and Thorpe discovered a symmetry of sectional curvatures which characterizes $4$-dimensional Einstein spaces. Later, this result was generalized to Einstein spaces of even dimensions $n = 2k \ge 4.$ We established curvature symmetries for Einstein spaces of arbitrary dimension $n \ge 4$.

### Einstein Metrics, Harmonic Forms, and Conformally Kaehler Geometry

There are certain compact 4-manifolds, such as real and complex hyperbolic 4-manifolds, 4-tori, and K3, where we completely understand the moduli space of Einstein metrics. But there are vast numbers of other 4-manifolds where we know that Einstein metrics exist, but cannot currently determine whether or not there are other Einstein metrics on them that are quite different from the currently-known ones. In this lecture, I will first present a characterization of the known Einstein metrics on Del Pezzo surfaces which I proved several years ago, and then describe an improved version which I obtained only quite recently.

### Blowing up extremal Poincaré type manifolds

Metrics of Poincaré type are Kähler metrics defined on the complement $X\setminus D$ of a smooth divisor $D$ in a compact Kähler manifold $X$ which near $D$ are modeled on the product of a smooth metric on $D$ with the standard cusp metric on a punctured disk in $\mathbb{C}$. In this talk I will discuss an Arezzo-Pacard type theorem for such metrics. A key feature is an obstruction which has no analogue in the compact case, coming from additional cokernel elements for the linearisation of the scalar curvature operator. I will discuss that even in the toric case, this gives an obstruction to blowing up fixed points (which is unobstructed in the compact case). This additional condition is conjecturally related to ensuring the metrics remain of Poincaré type.

### Big jet-bundles on resolution of orbifold surfaces of general type.

The presence of symmetric and more generally $k$-jet differentials on surfaces $X$ of general type play an important role in constraining the presence of entire curves (nonconstant holomorphic maps from $\mathbb{C}$ to $X$). Green-Griffiths-Lang conjecture and Kobayashi conjecture are the pillars of the theory of constraints on the existence of entire curves on varieties of general type.

When the surface as a low ratio $c_1^2/c_2$ a simple application of Riemann-Roch is unable to guarantee abundance of symmetric or $k$-jet differentials.

This talk gives an approach to show abundance on resolutions of hypersurfaces in $P^3$ with $A_n$ singularities and of low degree (low $c_1^2/c_2$). A new ingredient from a recent work with my student Michael Weiss gives that there are such hypersurfaces of degree $8$ (and potentially $7$). The best known result till date was degree $13$.

### Complete extremal metrics and stability of pairs on Hirzebruch surfaces

In this talk I will discuss the existence of complete extremal metrics on the complement of simple normal crossings divisors in compact Kähler manifolds, and stability of pairs, in the toric case.

Using constructions of Legendre and Apostolov-Calderbank-Gauduchon, we completely characterize when this holds for Hirzebruch surfaces. In particular, our results show that relative stability of a pair and the existence of extremal Poincaré type/cusp metrics do not coincide. However, stability is equivalent to the existence of a complete extremal metric on the complement of the divisor in our examples. It is the Poincaré type condition on the asymptotics of the extremal metric that fails in general.

This is joint work with Vestislav Apostolov and Lars Sektnan.

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Current organizer: Rosa Sena Dias