Abstracts

Romina Arroyo (Queensland)
On the signature of the Ricci curvature on nilmanifolds

One of the most important challenges of Riemannian geometry is to understand the Ricci curvature. A problem that is still open is: determine all possible signatures of the Ricci curvature of all Riemannian metrics on a given manifold. The aim of this talk is to present the problem in the setting of nilpotent Lie groups with left-invariant metrics, and give an answer in the case that the nilpotent Lie group admits a nice basis. This talk is based on work in progress with Ramiro Lafuente (The University of Queensland).

Renato Bettiol (CUNY)
Convex Algebraic Geometry of Curvature Operators

In this talk, I will discuss the structure of the set of algebraic curvature operators of n-dimensional manifolds satisfying a sectional curvature bound (e.g., nonnegative sectional curvature), under the light of the emerging field of Convex Algebraic Geometry. More precisely, we completely determine in what dimensions n this convex semi-algebraic set is a spectrahedron or a spectrahedral shadow (these are generalizations of polyhedra where linear programing extends to as semidefinite programming, which are of great interest in applied mathematics and optimization theory). Furthermore, for n=4, we describe the algebraic boundary of this set as the zero set of an explicit irreducible polynomial. This is based on joint work with M. Kummer (TU Berlin) and R. Mendes (Univ zu Köln).

Owen Dearricott (Melbourne)
Positive curvature in dimension seven

I will detail a general construction for putting positive sectional curvature on a series of 7-manifolds discussed as viable candidates to carry positively curved metrics with co-dimension one principal orbits discussed by Grove, Wilking and Ziller in JDG in 2008 by appropriately modifying a naturally occurring 3-Sasakian metric. This builds on previous work where I considered isolated examples of this sort where the base Bianchi IX self-dual Einstein orbifold metric had an algebraic closed form.

Anand Dessai (Fribourg)
Moduli spaces of $\sec \geq 0$ metrics in dimension $4k+1$

We will discuss recent results on the topology of moduli spaces of metrics of nonnegative sectional curvature on closed manifolds.

Joel Fine (Brussels)
Einstein 4-manifolds, negative curvature and smoothing cones

I will describe joint work with Bruno Premoselli which gives a new existence theorem for negatively curved Einstein $4$-manioflds, which are obtained by smoothing the singularities of hyperbolic cone metrics. Let $(M_k)$ be a sequence of compact $4$-manifolds and let $g_k$ be a hyperbolic cone metric on $M_k$ with cone angle $\alpha$ (independent of $k$) along a smooth surface $S_k$. We make the following assumptions:

• The injectivity radius $i(k)$ of $M_k$ tends to infinity (where in defining injectivity radius we ignore those geodesics which hit the cone singularity).
• The normal injectivity radius of $S_k$ is at least $i(k)/2$.
• The area of the singular locii satisfy $A(S_k)\leq C \exp(5 i(k)/2)$ for some $C$ independent of $k$.

Many examples of such sequences $(M_k,S_k)$ can be found using arithmetic constructions of hyperbolic manifolds. When these assumptions hold, we prove that for all large $k$, $M_k$ carries a smooth Einstein metric of negative curvature. The proof involves a gluing theorem and a parameter dependent implicit function theorem (where $k$ is the parameter). As I will explain, negative curvature plays an essential role in the proof. (For those who may be aware of our arxiv preprint, the work I will describe has a new feature, namely we now treat all cone angles, and not just those which are greater than $2\pi$. This gives lots more examples of negatively curved Einstein $4$-manifolds.)

David Gonzaléz Álvaro (Fribourg)
Open manifolds with positively curved souls

As Cheeger and Gromoll showed, an open manifold $N$ with a metric of non-negative (sectional) curvature contains a totally convex closed submanifold, called the soul, such that the total space of its normal bundle is diffeomorphic to $N$. For a given metric, the soul might not be unique but any two must be isometric. However, different metrics on the same open manifold may have "different" souls; we will review previous examples and we will construct open manifolds with non-homeomorphic positively curved souls. To do that we will study some properties of Eschenburg spaces, an infinite family of positively curved manifolds. This is joint work with Marcus Zibrowius.

Claudio Gorodski (São Paulo)
Actions on positively curved manifolds and boundary in the orbit space

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit space has non-empty boundary in the sense of Alexandrov geometry and prove that for sufficiently large $n$ (in terms of the Lie group), either the group has a fixed point, or a normal subgroup thereof, containing all isotropy groups associated to codimension one strata, has a fixed point. Among our applications, we classify representations of simple Lie groups whose orbit space has non-empty boundary. (Based on joint work with A. Kollross and B. Wilking.)

Karsten Grove (Notre Dame)
Almost non-negative curvature and rational ellipticity in cohomogeneity two

A fundamental conjecture by R. Bott suggests that all simply connected non-negatively curved manifolds $M$ are rationally elliptic, i.e., all but finitely many homotopy groups of such $M$ are finite. We will discuss an extension of this conjecture and a proof of it when in addition $M$ supports an isometric action with orbits of codimension at most two. This is joint work with Burkhard Wilking and Joseph Yeager.

Bernhard Hanke (Augsburg)
Local flexibility and counterintuitive approximations

In his famous 1986 monograph Gromov formulates a problem concerning local deformations of solutions to open partial differential relations. I will sketch a proof of the optimal result, which is based on the concept of generalized tangent spaces along arbitrary closed subsets of smooth manifolds. A number of applications will be presented, including the construction of Riemannian metrics with unexpected curvature properties on arbitrary manifolds. Joint work with Christian Bär.

John Harvey (Swansea)
Estimating the reach of a submanifold

The reach is an important geometric invariant of submanifolds of Euclidean space. It is a real-valued global invariant incorporating information about the second fundamental form of the embedding and the location of the first critical point of the distance from the submanifold. In the subject of geometric inference, the reach plays a crucial role. I will give a new method of estimating the reach of a submanifold, developed jointly with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar.

Robert Haslhofer (Toronto)
Ancient low entropy flows and the mean convex neighborhood conjecture

In this talk, I will explain our recent proof of the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb R^3$. Namely, if the flow has a cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\epsilon=\epsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\epsilon$ around $X$. The major difficulty is to promote the infinitesimal information captured by the tangent flow to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. As an application, we prove the uniqueness conjecture for mean curvature flow through cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed. This is joint work with Kyeongsu Choi and Or Hershkovits.

Panagiotis Konstantis (Stuttgart)
On the diffeomorphism type of GKM $6$-manifolds and application to Hamiltonian non-Kähler actions

We show that the diffeomorphism type of a $6$-dimensional GKM manifold is entirely encoded in its GKM graph. As an application we prove that the examples of Tolman and Woodward of manifolds with $T^2$ Hamiltonian non-Kähler actions are diffeomorphic to the Eschenburg flag $SU(3) // T^2$. This answers a question of Tolman on the existence of a (non-invariant) Kähler metric on her examples. Finally we will provide some properties of this Kähler metric. This is joint work with Oliver Goertsches and Leopold Zoller.

Brett Kotschwar (Arizona State)
On the maximal rate of convergence of the Ricci flow

We will show that solutions to the normalized Ricci flow which converge, up to diffeomorphisms, to an Einstein or soliton limit, either do so at an at-most exponential rate or are themselves self-similar. We will further discuss a connection between this unique-continuation result and the classification problem for shrinking Ricci solitons, and the application of the underlying method to other geometric flows.

Ramiro Lafuente (Queensland)
Homogeneous Einstein manifolds via a cohomogeneity-one approach

We establish non-existence results on non-compact homogeneous Einstein manifolds. The key idea in the proof is to consider non-transitive group actions on these spaces (more precisely, actions with cohomogeneity one), and to find geometric monotone quantities for the ODE that results from writing the Einstein equation in such a setting. As an application, we show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. This is joint work with C. Böhm.

Davi Maximo (UPenn)
On the topology and index of minimal surfaces

In this talk, I will discuss recent estimates relating the topology and index of minimal surfaces.

Jesús Núñez-Zimbrón (UNAM)
Maximal volume entropy of metric measure spaces with Ricci curvature bounded below

The volume entropy $h(M,g)$ of a compact, $n$-dimensional Riemannian manifold $(M,g)$ is a geometric invariant that measures the rate of exponential growth of the volume of concentric balls in the universal cover of $M$. This invariant is related to several other invariants such as the simplicial volume and minimal volume. Ledrappier and Wang showed that $h(M,g)$ is less than or equal to $n-1$ and that the equality holds if and only if $(M,g)$ is a hyperbolic manifold.

In this talk I will describe a generalization of this result to the class of metric measure spaces which have “Ricci curvature bounded below by $K$ and dimension bounded above by $N$” in the sense of the reduced Riemannian Curvature-Dimension condition ($RCD^*(K,N)$). This result is part of a joint project with C. Connell, X. Dai, R. Perales, P. Suárez-Serrato and G. Wei.

Jaime Santos Rodríguez (Madrid)
On Wasserstein isometries of closed Riemannian manifolds.

Let $\mathbb{P}_2(M)$ be the space of probability measures on a Riemannian manifold $M.$ Using optimal mass transport it is possible to define a distance on $\mathbb{P}_2(M),$ the so called $L^2-$Wasserstein distance $\mathbb{W}_2.$

In this talk we will discuss some intrinsic properties of Wasserstein spaces, more precisely we give a positive answer to the following question:
If two closed Riemannian manifolds $M, N$ are such that their corresponding Wasserstein spaces $\mathbb{P}_2(M),$ $\mathbb{P}_2(N)$ are isometric, does it follow then that $M$ is isometric to $N$?

Furthermore, in the case where $M=\mathbb{S}^n$ with its standard metric we show that the isometries of $\mathbb{P}_2(\mathbb{S}^n)$ must be of the form $g_{\#},$ where $g \in \mathrm{Isom}(\mathbb{S}^n).$ This differs from the case $M=\mathbb{R}^n,$ where Kloeckner proved that there exist some isometries of $\mathbb{P}_2(\mathbb{R}^n)$ that are not induced by Euclidean isometries. This is joint work with Prof. Luis Guijarro.

Lorenz Schwachhöfer (Dortmund)
Formality obstructions for highly connected manifolds

This talk will be based on the preprint arXiv:1902.08406 with D. Fiorenza, K. Kotaro and H.V. Lê.

The rational homotopy type of a closed oriented manifold $M$ is determined by the equivalence class of its deRham algebra $\Omega^\ast(M)$ w.r.t. quasi-isomorphisms of differential graded commutative algebras (DGCAs). If $\Omega^\ast(M)$ is quasi-isomorphically equivalent to its cohomology algebra $H^\ast(M)$, then $M$ is called formal.

In 2015, Crowley and Nordström introduced the Bianchi-Massey-tensor for $(r-1)$-connected $(r > 1)$ manifolds of dimension $n \leq 5r-3$ whose vanishing is equivalent to formality. As we can show, this tensor is equivalent to a class in Harrison cohomology and hence determines an $A_3$-algebra.

In this talk, I will explain how a simply connected Poincaré DGCA of Hodge type (a class which includes $\Omega^\ast(M)$ for $M$ closed simply connected) is equivalent to a finite dimensional DGCA, called its small quotient. In particular, if $M$ is $(r-1)$-connected $(r > 1)$ of dimension $n \leq 4r-1$, then the differential of the small quotient vanishes in all but one degree, whence any such algebra is almost formal. For instance, any closed simply connected $7$-manifold is almost formal (this was shown for $G_2$-manifolds by Chan, Karigiannis and Tsang last year). Furthermore, the small quotient in case of an $(r-1)$-connected $(r > 1)$ manifold of dimension $n \leq 5r-3$ gives an explicit description of the Bianchi-Massey tensor.

Catherine Searle (Wichita State)
Odd-dimensional nonnegatively curved GKM-manifolds

We prove that for odd-dimensional, closed, connected, orientable, non-negatively curved GKM$_3$ manifolds both the equivariant and the ordinary real cohomology split off the cohomology of an odd-dimensional sphere. This is joint work with Christine Escher and Oliver Goertsches.

Rafael Torres (SISSA)
Some new examples of $C^0$-nonnegatively curved involutions on $S^2\times S^{n - 2}$

We describe a construction of $C^0$-Riemannian metrics of nonnegative sectional curvature on orbit spaces of non-linear involutions on $S^2\times S^{n - 2}$, which do not arise from several of the known constructions of nonnegatively curved metrics. The orbit spaces include homotopy classes that were not previously known to contain nonnegatively curved manifolds.

Fred Wilhelm (Riverside)
Stability, finiteness and dimension 4

I will discuss the history and proof of the following result.
Theorem. For any $k \in \mathbb R$, $v >0$ and $D >0$, there are only finitely many diffeomorphism types of Riemannian $4$-manifolds with sectional curvature $\geq k$, volume $\geq v$ and diameter $\leq D$.
This is joint work in progress with Curtis Pro.

Rudolf Zeidler (Münster)
Band width estimates via the Dirac operator

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $\sigma > 0$, the distance between the boundary components of $V$ is at most $C/\sqrt{\sigma}$, where $C < 8 + 4\pi$ is a universal constant. This verifies a conjecture of Gromov for such manifolds.