Differential Geometry days @ UQ
Meeting on differential geometry at UQ, with a focus on geometric analysis and symmetries. There will be also a mini-course by Jorge Lauret in the previous days.
Dates Thursday 2 March (afternoon) - Friday 3 March, 2023
Social dinner Friday 3 March, 6:30pm @ Frida Kahlo Summer House.
Mini-course dates 22 February - 1 March (see information below)
Schedule
(All talks will be in room 67-442, 40 min + 10 min for questions.)
Thursday 2 March
2 - 2:50pm Christoph Böhm (Münster) "Compact homogeneous Einstein manifolds"
3 - 3:40pm Tim Buttsworth (UQ) "Cohomogeneity one ancient Ricci flows"
3:40 - 4:20pm break
4:20 - 5pm Owen Dearricott (La Trobe) "Formulae for self-dual Einstein metrics in terms of radicals"
Friday 3 March
9 - 9:40am James Stanfield (UQ) "Recent Progress on the curvatures of Gauduchon connections"
9:50 - 10:20am break
10:20 - 11am Elia Fusi (Torino) "On the Form-Type Calabi-Yau equation on 1-dimensional holomorphic bundles."
11:10 - 11:50am Kyle Broder (UQ) "Curvature aspects of hyperbolicity in complex geometry"
12 - 2pm lunch
2 - 2:40pm Yuri Nikolayevsky (La Trobe) "Geodesic orbit pseudo Riemannian nilmanifolds"
2:50 - 3:30pm Adam Thompson (UQ) "Cohomogeneity-one Ricci solitons with nilpotent symmetry"
3:30 - 4pm break
4 - 4:40pm Jorge Lauret (Córdoba) "Homogeneous generalized Einstein metrics"
6:30pm dinner
Abstracts
Christoph Böhm (Münster) "Compact homogeneous Einstein manifolds"
We will report on recent progress concerning the classification of homogeneous Einstein manifolds. If the Einstein constant for such a Riemannian metric is negative, then in recent joint work with R. Lafuente we could show that the underlying manifold must be diffeomorphic to a Euclidean space, confirming the famous Alekseevskii conjecture from 1975. In the Ricci flat case it is a classical result that such homogeneous Einstein metrics must be flat. The case when the Einstein constant is positive is wide open, even though there exist general existence and non-existence results.
Tim Buttsworth (UQ) "Cohomogeneity one ancient Ricci flows"
The construction and classification of ancient solutions to the Ricci flow is a popular subject in geometric analysis which has arisen out of the need to provide useful models of finite-time singularities of Ricci flow. In this talk, I will discuss classical examples of ancient Ricci flows which are rotationally-invariant, as well as some more recent constructions of ancient flows on spheres which are invariant under a product of two orthogonal groups.
Owen Dearricott (La Trobe) "Formulae for self-dual Einstein metrics in terms of radicals"
In the 90s Hitchin wrote down formulae for a handful of cohomogeneity one anti-self dual Einstein metrics in terms of rational functions. These were arrived at by exploiting a relationship between Cayley's criterion to characterise pairs of conics obeying Poncelet's porism and solutions to the Painlevé VI equation that in turn defined a cohomogeneity one ASD Einstein metric by work of Tod.
In this talk we discuss some important cases that are not well fleshed out in the literature, such as the solution of Painlevé VI associated with the Poncelet porism where the inscribing-circumscribing polygons have an even number of sides.
Moreover, we provide some explicit metrics with neutral signature and others with unusual cone angle singularities along a singular real projective plane that were speculated about by Atiyah and LeBrun.
James Stanfield (UQ) "Recent Progress on the curvatures of Gauduchon connections"
On a generic Hermitian manifold, the complex structure is not compatible with the Levi-Civita connection. Instead, one considers metric connections that are compatible with the complex structure. The space of such connections is in general infinite dimensional. In the 90's, Gauduchon introduced a "canonical" one dimensional subspace which included all previously distinguished Hermitian connections (in particular, the Chern and Bismut connections). In this talk, we will discuss some recent results regarding the curvature properties of these connections. In particular, we will present the classification of compact Hermitian manifolds with flat Gauduchon connection by confirming a conjecture of Yang and Zheng. Time permitting, we will also discuss some recent results on the Ricci curvatures and Holomorphic sectional curvatures associated with these connections.
Elia Fusi (Torino) "On the Form-Type Calabi-Yau equation on 1-dimensional holomorphic bundles."
The Form-Type Calabi-Yau equation was introduced by Fu, Wang and Wu to attack the so-called dilatino equation of the Hull-Strominger system.
In this talk, after briefly introducing the Hull-Strominger system and the general formulation of the Form-Type Calabi-Yau equation, we will recall some of the main results concerning said equation. Afterwards, we will introduce our ansatz, firstly in the case of complex parallelizable nilmanifolds, then in the case of holomorphic bundles with 1-dimensional fibers, describing the results obtained so far.
This is a joint work in progress with Luigi Vezzoni.
Kyle Broder (UQ) "Curvature aspects of hyperbolicity in complex geometry"
A compact complex manifold X is said to be Kobayashi hyperbolic if every holomorphic map from the complex plane to X is constant. An extension of a conjecture of Kobayashi predicts that all compact Kobayashi hyperbolic manifolds are projective and admit a Kähler--Einstein metric with negative Ricci curvature. We will discuss the curvature aspects of Kobayashi hyperbolic manifolds and their study via Gauduchon connections. We will also present a "positive analog" of the Kobayashi conjecture.
Yuri Nikolayevsky (La Trobe) "Geodesic orbit pseudo Riemannian nilmanifolds"
We know that in the Riemannian case, (i) for every homogeneous space, there is a reductive decomposition at the level of Lie algebras, (ii) the isometry group of a simply connected nilmanifold is the semidirect product of isometric automorphisms and translations (Wolf/Wilson), and (iii) geodesic orbit nilmanifolds are necessarily two-step nilpotent or abelian (Gordon). Neither of this is true in pseudo-Riemannian signature. However, it turns out that in low signature, some results may still be “rescued”. This is a joint work (which is partially still in progress) with Joe Wolf, Zhiqi Chen and Shaoxiang Zhang.
Adam Thompson (UQ) "Cohomogeneity-one Ricci solitons with nilpotent symmetry"
There are many examples of Ricci solitons that are constructed using the following ansatz: the soliton admits a cohomogeneity-one group action by a compact Lie group. On the other hand, there are very few examples of cohomogeneity-one Ricci solitons where the group acting is non-compact. We will discuss our construction of new examples of complete cohomogeneity-one gradient Ricci solitons where the group action is by a simply connected Nilpotent Lie group.
Jorge Lauret* (Córdoba) "Homogeneous generalized Einstein metrics"
A generalized metric on a manifold M, i.e., a pair (g,H), where g is a Riemannian metric and H is a closed 3-form, is a fixed point of the generalized Ricci flow if and only if (g,H) is Bismut Ricci flat (or generalized Einstein): H is g-harmonic and Ric(g) is one fourth of the square of H w.r.t. g. On any homogeneous space M=G/K, where G is a compact semisimple Lie group with two simple factors G_1 and G_2, under some mild assumptions, we give a G-invariant generalized Einstein metric. Moreover, if K is semisimple and the standard metric is Einstein on both G_1/K_1 and G_2/K_2, where K_1 and K_2 are the projections of K, we provide a one-parameter family of such structures, which is shown to be most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for M=SO(8)xSU(4)/SU(3) and for every M=GxG/Diag K, where G/K is an irreducible symmetric space with K simple.
Mini-course: The Ricci curvature of compact homogeneous spaces
All talks in 67-442
Schedule
Lec 1 Wednesday 22 February, 11am - noon
Formulas for the Ricci and scalar curvatures, the Ricci map, the Hessian of the scalar curvature functional Sc and the Lichnerowicz Laplacian. Application to the Prescribed Ricci Curvature Problem.
Lec 2 Friday 24 February, 11am - noon
Stability of Einstein metrics as critical points of Sc. Application to generalized Wallach spaces and flag manifolds.
Lec 3 Monday 27 February, 2 - 3pm
Stability of standard Einstein manifolds. New examples of local maxima of Sc.
Lec 4 Wednesday 1 March, 11am - noon
Generalized Einstein metrics. Cohomology of compact homogeneous spaces and harmonic 3-forms.