Symmetry and shape

Ilka Agricola (Philipps-Universität Marburg, Germany)

Dmitri Alekseevsky (Higher School of Modern Mathematics MIPT, Russia)

Contact hypersurfaces in Hermitian symmetric spaces

Jürgen Berndt (King's College London, United Kingdom)

Masahiro Kon classified in 1979 the contact hypersurfaces in complex projective spaces. In the talk I will discuss the classification problem for contact hypersurfaces in Hermitian symmetric spaces of compact type.

Bibliography:

1. M. Kon, Pseudo-Einstein real hypersurfaces in complex projective spaces, J. Differ. Geom. 14 (1979), 339-354.

Antonio Di Scala (Politecnico di Torino, Italy)

Anna Fino (Università di Torino, Italy)

Claudio Gorodski (Universidade de São Paulo, Brazil)

Luis Guijarro (Universidad Autónoma de Madrid, Spain)

Jason Lotay (University of Oxford, United Kingdom)

Paolo Piccione (Universidade de São Paulo, Brazil)

Silvio Reggiani (Universidad Nacional de Rosario, Argentina)

Alberto Rodríguez Vázquez (Université Libre de Bruxelles, Belgium)

Evangelia Samiou (University of Cyprus, Cyprus)

Francisco Vittone (Universidad Nacional de Rosario, Argentina)

Equivariant Approximation Process of Currents and Spaces with Bounded Curvature

Andrés Ahumada Gómez (CUNEF Universidad, Spain)

In this talk (poster) we introduce actions of a compact Lie group in two regularization processes: in De Rham's approximation process of currents on a smooth manifold by smooth currents, and in a smoothing operator of Riemannian metrics of metric spaces with bounded curvature.

Biquotients that are orbifolds

Nícolas Roberto Ribeiro Caballero (University of São Paulo, Brazil)

Biquotients are generalizations of homogeneous spaces. The foundations of the theory trace back to Eschenburg's Habilitation [1] in 1984, which was motivated by the representation of an exotic 7-sphere as a biquotient by Gromoll and Meyer [4].

More precisely, let $G$ be a Lie group with a left-invariant metric, and let $H$ be a subgroup of $G \times G$. Denote by $H_l$ and $H_r$ the projections of $H$ onto the left and right factors, respectively, and assume that the metric on $G$ is $H_r$-invariant. Then the action of $H$ on $G$, \[ (h_l,h_r) \cdot g = h_l g h_r^{-1}, \] with $(h_l, h_r) \in H$, is isometric. The quotient space of this action is denoted by $G//H$ and is called a biquotient. If the action of $H$ on $G$ is (respectively, almost) free, then $G//H$ is a manifold (respectively, an orbifold), and a quotient Riemannian metric is defined such that the projection $G \to G//H$ is a Riemannian submersion. Recall that a Riemannian orbifold of dimension $n$ is a length space locally isometric to the quotient of an $n$-dimensional Riemannian manifold by a finite group of isometries.

The theorem of Lytchak and Thorbergsson [6] gives a necessary and sufficient condition for a biquotient to be an orbifold. In the context of biquotients, the theorem states that $G//H$ is an orbifold if and only if all slice representations of the $H$-action on $G$ are polar. In other words, $G//H$ is a biquotient orbifold if and only if the $H$-action is infinitesimally polar. This characterization has already been used in other classifications of biquotients, among which we mention the biquotients generated by spheres studied by Gorodski and Lytchak [2], and those in compact rank-one symmetric spaces, by Gorodski and Kollross [3].

In this doctoral project, we are classifying infinitesimally polar actions on compact Lie groups of low rank, in order to identify which biquotients admit an orbifold structure.

Bibliography:

1. J. H. Eschenburg, Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenr. Math. Inst. Univ. Münster 32 (1984).
2. C. Gorodski, A. Lytchak, Isometric actions on spheres with an orbifold quotient, Math. Ann. 365 (2016), 1041-1067.
3. C. Gorodski, A. Kollross, Some remarks on polar actions, Ann. Glob. Anal. Geom. 49 (2016), 43-58.
4. D. Gromoll, W. Meyer, An Exotic Sphere With Nonnegative Sectional Curvature, Ann. of Math. 100(2) (1974), 401-406.
5. A. Kollross, A classification of hyperpolar and cohomogeneity one actions, Trans. Amer. Math. Soc. 354(2) (2002), 571-612.
6. A. Lytchak, G. Thorbergsson, Curvature explosion in quotients and applications, J. Differential Geom. 85(1) (2010), 117-139.

Spectrum of Kähler manifolds

Sayantan Chakraborty (Michigan State University, India)

The results of Lichnerowicz and Obata give optimal lower bounds for the first eigenvalue on Riemannian manifolds with positive Ricci lower bound as well as rigidity when equality is achieved. Kähler manifolds are special class of Riemannian manifolds where three structures on the manifold interact in a compatible way. I will discuss Lichnerowicz and Obata type theorems for closed Kähler manifolds and then for compact Kähler manifolds with convex boundary (and both with Ricci lower bound conditions). We will see that if equality is achieved in the case of compact Kähler manifolds with positive Ricci lower bound and convex boundary, the boundary is totally geodesic and there is a nontrivial holomorphic vector field on the manifold. If time permits, I will talk about multiplicity of eigenvalues on Kähler manifolds and mention some related results.

Index of symmetry of 2-isotropy irreducible homogeneous spaces

Ángel Cidre Díaz (Universidade de Santiago de Compostela, Spain)

Symmetric spaces are among the best-understood classes of Riemannian manifolds and can be seen as a natural generalization of space forms. The index of symmetry is a geometric invariant that, in a certain sense, measures how far a Riemannian metric is from being that of a symmetric space, since it attains its maximum value (equal to the dimension of the manifold) if and only if the manifold is a locally symmetric space. The index and co-index of symmetry have been mainly studied in the context of compact, irreducible, and simply connected homogeneous Riemannian manifolds, where several general structural results have been obtained.

Among compact homogeneous spaces, it is known that when the isotropy representation is irreducible, the index of symmetry is either zero or maximal. A particular case of such spaces is given by symmetric spaces, when viewed as quotients by the identity component of their isometry group. Thus, studying the index of symmetry in isotropy irreducible homogeneous spaces essentially reduces to the well-known theory of symmetric pairs. This naturally raises the question of what happens when the space is no longer isotropy irreducible.

In this talk, we explore the simplest nontrivial case: when the isotropy representation decomposes into exactly two irreducible modules. These spaces are known as 2-isotropy irreducible homogeneous spaces, providing a natural next step beyond the classical isotropy irreducible case.

This is joint work with Carlos Enrique Olmos (Universidad Nacional de Córdoba) and Alberto Rodríguez Vázquez (Université Libre de Bruxelles).

Bycicle tracks with hyperbolic monodromy

Luis Hernández-Lamoneda (Centro de Investigación en Matemáticas, A.C., México)

We find new necessary and sufficient conditions for the bicycling monodromy of a closed plane curve to be hyperbolic. The manin tool is the "hyperbolic development" of a euclidean plane curve in the hyperbolic plane.

Towards a Ricci flow with surgery from Taub-Bolt to Taub-NUT

John Hughes (University of Oxford, United Kingdom)

A conjecture of Holzegel, Schmelzer and Warnick states that there is a Ricci flow with surgery connecting the two Ricci flat metrics Taub-Bolt and Taub-NUT. We will present some recent progress towards proving this conjecture.

Complete cohomogeneity one hypersurfaces into the hyperbolic space

Fernando Manfio (University of São Paulo, Brazil)

In this talk we will consider isometric immersions $f\colon M^n\to\mathbb{H}^{n+1}$ into hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of dimension $n$ on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We give a complete classification if either $n\geq 3$ and $M^n$ is compact or if $n \geq 5$ and the connected components of the flat part of $M^n$ are bounded. This is a joint work with Felippe Guimarães and Carlos Olmos.

Conformal transformations between weighted Einstein manifolds

Diego Mojón Álvarez (Universidade de Santiago de Compostela, Spain)

A classical problem in Riemannian geometry is that of determining which manifolds admit multiple Einstein representatives of the same conformal class. In the setting of smooth metric measure spaces (Riemannian manifolds endowed with a smooth density function), many Riemannian invariants are naturally modified so that they incorporate information on both the density and the metric. This process gives rise to weighted analogues of Einstein manifolds and conformal classes, allowing for the classical problem to be reformulated in a weighted context.

We analyze smooth metric measure spaces admitting two weighted Einstein structures in the same weighted conformal class, completing partial results found in the literature [2], [3]. We begin by describing the local geometries of these manifolds in terms of certain Einstein and quasi-Einstein warped products, as well as the forms of the density function and the conformal factor relating both structures. Subsequently, we obtain a global classification result under the assumption that one of the underlying metrics is complete. We show that such a manifold is either a weighted analogue of a space form or it belongs to specific families of Einstein or quasi-Einstein warped products. As a consequence, in the compact case, it must be a weighted round sphere.

Bibliography:

1. M. Brozos-Vázquez, E. García-Río, D. Mojón-Álvarez, Conformally weighted Einstein manifolds: the uniqueness problem, arXiv:2504.07860 [math.DG].
2. J. S. Case, The weighted $\sigma_k$-curvature of a smooth metric measure space, Pacific. J. Math. 299 (2) (2019), 339-399.
3. J. S. Case, A Yamabe-type problem on smooth metric measure spaces, J. Differential Geom., 101 (3) (2015), 467-505.

Killing tensors on symmetric spaces

Yuri Nikolayevsky (La Trobe University, Melbourne, Australia)

I will present some old and new results on geometry and algebra of higher rank Killing tensor fields on Riemannian symmetric space, with a particular focus on the following question: "Is any Killing tensor field on a symmetric space a polynomial in Killing vector fields?" The talk will include some joint results with Vladimir Matveev (Germany) and An Ky Nguyen (Australia).

Cohomogeneity one actions on symmetric spaces of mixed type

Tomás Otero Casal (Universität Münster, Germany)

I will report on ongoing work with Ivan Solonenko and Hiroshi Tamaru where we study cohomogeneity one actions on (products of) symmetric spaces. We show that, with the only exception of a $2$-parameter family of diagonal actions, cohomogeneity one actions on a given symmetric space $M_+\times M_0\times M_-$ (where $M_+$ is a symmetric space of compact type, $M_0$ a euclidean space, and $M_-$ is a symmetric space of noncompact type) decompose as products of actions on each of the individual factors. In particular, we give an explicit description of homogeneous codimension one foliations for any given simply-connected symmetric space.

Curvature-adapted orbits of CH1 actions in symmetric spaces of non-compact type

Mario Julián Rodríguez Sánchez de Toca (Universidade de Santiago de Compostela, Spain)

In Riemannian Geometry, two of the most important operators when studying submanifolds are the shape operator and the Jacobi operator. A hypersurface is said to be curvature-adapted precisely when these two operators have a nice behavior between them, that is, when they diagonalize simultaneously. In this talk we will investigate this property in the presence of symmetry. More precisely, we will provide the classification of curvature-adapted homogeneous hypersurfaces in symmetric spaces of non-compact type.

Ricci Flow on Five-Dimensional Homogeneous Manifolds

Daniel Rotmeister Teixeira de Barros (IME-USP, Brazil)

In 1982, Richard Hamilton introduced a geometric flow derived from an evolution equation involving a Riemannian metric and the respective Ricci tensor. He named this flow by Ricci flow and it can be regarded as a non-linear version of the heat equation. This means that the Ricci flow induces a diffusion of the curvature on the manifold. Since the seminal paper of R. Hamilton [3], the Ricci flow has been intensively studied, revealing interesting connections between geometry and topology. A proficuous application of the Ricci flow was provided by Grigori Perelman in order to prove the celebrated Thurston Geometrization Conjecture, of which the Poincaré Conjecture is a consequence.

The aim of this work is to analyze the Ricci flow on five-dimensional homogeneous manifolds in the sense of Thurston (manifolds equipped with an invariant Riemannian metric and admitting compact quotients), as recently classified by Andrew Geng in [2]. Some topics to be explored include the possible pointed limits of the Ricci flow under several normalizations, the existence of ancient or immortal solutions, and the classification of singularity types. Our approach is based on an equivalent flow, namely the bracket flow, introduced by Jorge Lauret in [4], [5], which explores the algebraic behavior of homogeneous manifolds. For example, in the case of Lie groups, instead of making the Riemannian metric evolve, we fix it and evolve the Lie bracket in the Lie algebra. The case of homogeneous spaces is more delicate and requires a reductive decomposition. Technically, this leads to a dynamical system defined on a subset of the algebraic variety of Lie brackets. In the literature, the case of dimensions 3 and 4 was analyzed by Jorge Lauret and Romina Arroyo in [5], [1], respectively.

We would like to thank FAPESP for financial support.

Bibliography:

1. R. M. Arroyo, Sobre el flujo de Ricci en variedades homogéneas, Ph.D. thesis, Universidad Nacional de Córdoba, 2013.
2. A. L. L. Geng, The classification of five-dimensional geometries, Ph.D. Thesis, The University of Chicago, 2016.
3. H. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306.
4. J. Lauret, Convergence of homogeneous manifolds, J. Lond. Math. Soc. (2) 86 (2012), no. 3, 701-727.
5. J. Lauret, Ricci flow of homogeneous manifolds, Math. Z. 274 (2013), no. 1-2, 373-403.

Nonsmooth Lorentzian geometry

Marta Sálamo Candal (University of Vienna, Austria)

In recent years, there have been many developments in the field of Lorentzian geometry motivated by the introduction of Lorentzian length spaces by Kunzinger-Sämann (2017). This led to a new formulation of the geometry of general relativity, providing a framework that allows the study of nonsmooth spacetimes, including those that present singularities or with a discrete structure. An important development, in particular, is the characterisation of Ricci curvature bounds by the convexity of an entropic functional, firstly developed by McCann (2018) and Mondino-Suhr (2018) in the smooth setting, and taken into the synthetic setting by Cavalletti-Mondino (2020). In this talk, I will review the basics of this new geometry for Einstein's theory of relativity, and I will discuss some of the research directions that are currently being developed.

Exterior Curvature in Hitchin's Generalised Geometry

Oskar Schiller (Universität Hamburg, Germany)

Generalised Geometry is concerned with the study of geometrical objects on the generalised tangent bundle $TM \oplus T^* M$ over a manifold $M$. The basic objects are generalised metrics, generalised connections, and divergence operators. From these, one can obtain other geometrical objects, such as the generalised Riemann tensor.

In this talk, which is based on joint work with Vicente Cortés, we consider an immersed manifold $N \hookrightarrow M$, and investigate how $TN \oplus T^*N$ inherits geometric structures defined on $TM \oplus T^*M$. Assuming $N$ is a hypersurface, we develop the notion of generalised exterior curvature, introducing the generalised second fundamental form and the generalised mean curvature. We present generalised versions of the Gauß-Codazzi equations, and discuss possible applications.

Classification of cohomogeneity-one actions on reducible symmetric spaces of compact type

Ivan Solonenko (University of Stuttgart, Germany)

I will report on a joint work in progress with Andreas Kollross, in which we are extending the classification of isometric cohomogeneity-one actions on irreducible symmetric spaces of compact type obtained by Kollross in 1998 to the general reducible case. In order to do so, we first obtain an explicit classification of isometric transitive actions on the aforementioned (reducible) spaces up to subgroup-conjugacy by using the works of Onishchik on decompositions of simple compact Lie algebras. I will restrict to the (less convoluted) case when all the de Rham factors are of non-group type and make comments on the general case.

Computer-assisted construction of cohomogeneity one Einstein metrics

Qiu Shi Wang (University of Oxford, United Kingdom)

In this talk, I will present a construction of a new family of $SU(2)$-invariant complete negative Einstein metrics on the complex line bundle $O(-4)$ over $\mathbb{C}P^1$. The proof involves using rigorous numerics to produce an approximate Einstein metric to high precision in a bounded region near the singular orbit, then perturbing it to a genuine Einstein metric using fixed-point methods. At the boundary of this region, we show that the latter metric is sufficiently close to hyperbolic space so that it extends to a complete, asymptotically hyperbolic Einstein metric. Our construction is based on previous work of Buttsworth-Hodgkinson.

Spheres with parallel mean curvature in $\mathbb{S}^2\times \mathbb{H}^2$

Giel Stas (KU Leuven, Belgium)

In Riemannian products of two real space forms, it is known that surfaces with parallel mean curvature vector admit a holomorphic quadratic differential. We consider the product of a sphere and a hyperbolic plane with opposite sectional curvature. Within this product space, we study the parallel mean curvature surfaces for which the differential vanishes. In particular, this includes the surfaces which have the topology of a sphere.