Abstracts
Christian Bär, Universität Potsdam
Boundary value problems for the Dirac operator on Riemannian and Lorentzian manifolds
The Atiyah-Patodi-Singer index theorem states that the Dirac operator on a compact Riemannian spin manifold subject to some particuar spectral boundary conditions is Fredholm and it provides a geometric formula for its index. I will first talk about joint work with W. Ballmann describing all boundary conditions giving rise to Fredholm operators. As a byproduct, I will give a very short proof of the relative index theorem by Gromov and Lawson.
In the second part I will explain index theory for the Dirac operator on Lorentzian manifolds and emphasize both the similarities and the differences to the Riemannian counterpart. This is joint work with A. Strohmaier and S. Hannes.
Renato Bettiol, University of Pennsylvania
Non-negative sectional curvature and Weitzenböck formulae
We show that an algebraic curvature operator has non-negative sectional curvature if and only if its curvature term in the Weitzenböck formulae for symmetric $k$-tensors is non-negative for all $k$. This equivalence is analogous to a result of Hitchin; that an algebraic curvature operator is positive-semidefinite if and only if its curvature term in the Weitzenböck formulae for all orthogonal representations is nonnegative. Similar algebraic characterizations for any lower/upper bounds on sectional curvature can be derived; as well as potential decision algorithms to determine whether a curvature operator satisfies such bounds. This is based on joint work with R. Mendes.
Christoph Böhm, Universität Münster
Homogeneous Ricci flows
We will report on new structure results for homogeneous Ricci
flows, but will also address (many) open questions.
Diego Corro Tapia, Karlsruher Institut für Technologie
Positive Ricci curvature on manifolds with large torus actions
For a simply-connected $(n+2)$-manifold $M$ with a smooth effective action of the $n$-torus we construct an invariant Riemannian metric with positive Ricci curvature.
Our work generalizes a construction of Bazaikin and Matvienko to all dimensions to get the following result:
Theorem. If $M$ is a smooth closed simply-connected $(n+2)$-manifold that admits an effective $n$-torus action, then $M$ admits an equivariant smooth Riemannian metric of positive Ricci curvature.
Using work of Oh, our construction yields examples in all dimensions, and for $M$ of dimension $4$, $5$, and $6$ an explicit list of manifolds can be given.
David González, Universidad Autónoma de Madrid
Non-negative sectional curvature on stable classes of vector bundles
In this talk we will discuss the following question, motivated by Cheeger-Gromoll's Soul Theorem: given a vector bundle $E$ over a compact manifold, does the product $E\times\mathbb{R}^k$ admit a metric with non-negative sectional curvature for some $k$? We will give an affirmative answer for every vector bundle over (almost) any homogeneous space with positive curvature. We will extend this result to include further classes of homogeneous spaces, and we will show that the question above is stable under tangential homotopy equivalences. This is joint work with Marcus Zibrowius.
Lee Kennard, University of Oklahoma
Positive curvature and torus symmetry
It is an open problem whether every simply connected, closed manifold admitting non-negative sectional curvature also admits positive sectional curvature. One conjectured obstruction is due to Hopf: In even dimensions, positive sectional curvature implies positive Euler characteristic. I will discuss joint work with Burkhard Wilking on this problem in the presence of torus symmetry.
Brett Kotschwar, Arizona State University
Asymptotic rigidity of noncompact shrinking gradient Ricci solitons
Shrinking gradient Ricci solitons are generalized fixed points of the Ricci flow equation and models for the geometry of a solution in the neighborhood of a developing singularity.
At present, all known examples of complete noncompact shrinkers are either asymptotic to a regular cone at infinity or are locally reducible as products, and there is growing evidence that, at least in four dimensions, these are the only possibilities. I will discuss some recent uniqueness results obtained in part with Lu Wang which demonstrate that a shrinker which is either smoothly asymptotic to a cone or to a generalized cylinder in an appropriate sense along some end is essentially determined by this asymptotic geometry.
Nina Lebedeva, Steklov Institute, St. Petersburg
Dipole curvature condition for metric spaces.
We introduce a new condition for (k+l+2)-point subsets in metric spaces and call it (k,l)-dipole property. For intrinsic metric space this condition is more strong then nonnegative curvature condition and holds naturally for quotients of Euclidean spaces by isometric group action. It holds also for biquotient spaces of Lie groups with bi-invariant metric.
We show that (3,1) and (2,2)-dipole properties hold for any CBB(0) space while (3,3)-dipole property doesn't. The first two give new conditions for 6-point subsets in CBB(0) space and the last gives a condition which allows to see the difference between subsets of CBB(0) space and subsets of quotients of Euclidean spaces.
We also prove that (4,1)-dipole property implies Ma-Trudinger-Wang (MTW) condition. MTW condition is a 4th-order
nonlocal curvature condition (more strong then nonnegative sectional curvature) related to the regularity of optimal transport. This gives a hope to prove continuity of optimal transport without using a smooth structure. As an application we note that biquotient spaces of Lie groups with bi-invariant metric satisfy MTW condition.
This is joint work with Anton Petrunin and Vladimir Zolotov.
Matthias Ludewig, Max-Planck-Institut für Mathematik, Bonn
The mass of conformal Laplacians
We introduce the notion of the mass of conformal powers of the Laplacian, which is a certain spectral invariant connected to the ADM mass of an associated asymptotically flat manifold. One of its main features is that in odd dimensions, the mass -- although it is a global invariant -- transforms locally under a conformal change. The mass also seems to capture several global geometric and topological features of the manifold. Here we discuss several recent results.
Alexander Lytchak, Universität zu Köln
Geodesic flow on singular spaces
In the talk I will discuss the proof of the following theorem about singular hypersurfaces obtained jointly with V. Kapovitch und A. Petrunin: For any convex hypersurface in the Euclidean space, almost every of its tangent directions is the starting direction of an infinite geodesic. The result will be a special case of a theorem relating the existence of a geodesic flow on Alexandrov spaces to a new metric-measure invariant, which has very close but rather intricate ties with integral geometry. This metric measure invariant seems to be of some interest beyond Alexandrov geometry.
Ricardo Mendes, Universität zu Köln
Minimal hypersurfaces in compact symmetric spaces
A conjecture attributed to R. Schoen states that if $(M,g)$ is a compact Riemannian manifold with positive Ricci curvature, then there exists $C>0$ such that any closed minimal hypersurface $S$ satisfies $\mathrm{\mathrm{index}}(S)>C \cdot b_1(S)$. Here $b_1$ denotes the first Betti number, and $\mathrm{index}(S)$ denotes the Morse index of $S$ for the area functional. In previous work (by Ros, Savo, Ambrozio-Carlotto-Sharp, ...) this conjecture has been established for $(M,g)$ any compact rank-one symmetric space (CROSS), with $g$ the standard metric. Moreover a weak version has been proved for (flat) tori, "weak" in the sense the index is replaced with the index plus nullity.
Our main result is such a weak version of Schoen's conjecture valid for any compact symmetric space. If the rank is one we also recover the strong version, albeit with a worse constant $C$ than in the previous results.
The main new tool used in our proof is a generalization of isometric immersions of $(M,g)$ into Euclidean space. Namely, we consider isometric embeddings of the tangent bundle $TM$ into a trivial vector bundle $M \times \mathbb R^n$, such that the standard connection on $M \times \mathbb R^n$ induces the Levi-Civita connection of $M$. The fundamentals of the theory of isometric immersions generalize in similar form, with the important caveat that the second fundamental form is no longer necessarily symmetric.
If time allows I will also discuss cases where a robust version of such index bounds holds, in the sense that there is a constant $C'$, and a neighbourhood $U$ of the metric $g$, such that for every metric $g'$ in $U$, and every closed minimal hypersurface $S$ of $(M,g')$, one has $\mathrm{index}(S)>C' \cdot b_1(S)$. These include the CROSS, the Lie groups $Sp(n)$, and the quaternionic Grassmannians. This is joint work with Marco Radeschi.
Hans-Bert Rademacher, Universität Leipzig
Critical values of homology classes of loops and positive curvature
We consider a Riemannian metric $g$ of positive sectional curvature $K \geq 1$ on the $n$-sphere $S^n$. Fix a point $p \in S^n$ and denote by $\Omega_p S^n$ the space of loops on $S^n$ with base point $p$. Morse theory assigns to the non-trivial homology class $h_p \in H_{n-1} (\Omega_p S^n )$ its critical value $L_p = \text{cr}(h_p )$ which equals the length $L_p = L(c_p )$ of a geodesic loop $c_p$ at which the homology class $h_p$ remains hanging. We show that $L_p \leq 2\pi$ and that equality characterizes the round metric. The same result holds for the critical value $L = \text{cr}(h)$ of the non-trivial homology class $h \in H_{n-1} (\Lambda S^n )$ in the free loop space $\Lambda S^n$. This critical value equals the length $L = L(c)$ of a closed geodesic $c$. Under the additional assumption $K \leq 4$ this follows from results by Sugimoto (Tôhoku Math. J. 22, 1970) in even dimensions and Ballmann, Thorbergsson & Ziller (Comment. Math. Helv. 58, 1983).
As a consequence one obtains that only for the round metric the length of a shortest closed geodesic attains its maximal value $2\pi$. This result was stated in unpublished preprints by Itokawa & Kobayashi (1991, 2005, 2008).
Marco Radeschi, University of Notre Dame
Odd dimensional Besse orbifolds are manifolds
Riemannian orbifolds enjoy most properties of Riemannian manifolds, and in particular the existence of geodesics. When all geodesics can be extended indefinitely and are all periodic, we say that the orbifold is a Besse orbifold. The goal of this talk is to show that, if a Besse orbifold has odd dimension, it must in fact be a manifold, in fact a sphere. This is a joint work in progress with Manuel Amann and Christian Lange.
Frank Reidegeld, Techinische Universität Dortmund
$G_2$-orbifolds with ADE-singularities
Let $M$ be a $7$-dimensional orbifold whose singularities are modeled on $\mathbb R^7/G$ where $G$ can be embedded into the exceptional group $G_2$. It is possible to define the notion of a $G_2$-structure on $M$. We study parallel $G_2$-structures on orbifolds with a special kind of singularities, namely ADE-singularities. Orbifolds of this kind have applications in M-theory and they may define boundary components of the moduli space of parallel $G_2$-structures. We show how the existing construction methods for $G_2$-manifolds can be modified such that they produce $G_2$-orbifolds. In addition, we obtain an example of a smooth $G_2$-manifold with new values of the Betti numbers $b_2$ and $b_3$.
Jaime Santos Rodríguez, Universidad Autónoma de Madrid
Isometries and Ricci curvature on metric measure spaces.
The notion of synthetic lower Ricci curvature bounds was independently defined for metric measure spaces, $(X,d,\mathfrak{m}),$ by Sturm and Lott-Villani. This condition, called curvature-dimension condition $CD(K,N),$ is given in terms of the optimal transport between probability measures on $X$ and the convexity of an entropy functional. It is known that a Riemannian manifold $(M^n, g)$ having Ricci curvature bounded below by $(n-1)K$ is equivalent to the metric measure space $(M^n,d_g, d\text{Vol}_g)$ being a $CD(K,N)$ space. This condition is also known to be stable under measured Gromov-Hausdorff convergence.
We say that $(X,d, \mathfrak{m})$ is infinitesimally Hilbertian, if the associated Sobolev space $W^{1,2}(X,d,\mathfrak{m})$ is a Hilbert space. A metric measure space satisfies the Riemannian Curvature Dimension condition, $RCD(K,N)$, if it is both infinitesimally Hilbertian and $CD(K,N)$.
This condition rules out Finsler geometries and provides good structural results such as an extension of the classical Splitting theorem of Cheeger-Gromoll done by Gigli.
In this talk we will discuss some properties and examples of $RCD(K,N)$ spaces and present results about the group of isometries of a $RCD(K,N)$ space being a Lie group. This is joint work with Luis Guijarro.
Catherine Searle, Wichita State University
The maximal symmetry rank conjecture for non-negatively curved manifolds
The maximal symmetry rank conjecture states:
Conjecture. Let $T^k$ act isometrically and effectively on $M^n$, a closed, simply-connected, non-negatively curved Riemannian manifold. Then
In particular, we have shown that for maximal and almost maximal torus actions of rank $ \lfloor 2n/3\rfloor$ that the conjecture holds. I'll discuss the proof of this result as well as some consequences regarding the classification of manifolds of non-negative curvature with maximal and almost maximal symmetry rank in low dimensions. This is joint work with Christine Escher.
Anna Siffert, Max-Planck-Institut für Mathematik, Bonn
Existence of metrics maximizing the first eigenvalue on closed surfaces
We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that maximizes the first eigenvalue of the Laplace operator among all unit volume metrics. The key ingredient are several monotonicity results, which have partially been conjectured to hold before. This is joint work with Henrik Matthiesen.
Llohann Sperança, Universidade Federal de São Paulo
Totally geodesic Riemannian foliations on compact Lie groups
In this work we deal with a natural question: how to fill a specific geometric space with a (locally) repeating pattern? If one thinks in decomposing a Riemannian manifold into locally equidistant submanifolds (called leaves), such decomposition is called a Riemannian foliation (foliations here are assumed non-singular).
For instance, given a compact Lie group $G$, any subgroup $H \subset G$ produces a foliation whose leaves are the translated copies of $H$. We call such foliations as homogeneous. If $G $ is endowed with a bi-invariant metric, one sees that homogeneous foliations are Riemannian and have totally geodesic leaves.
Here we prove the converse of this statement: let $G$ be a compact Lie group with a bi-invariant metric. Then a Riemannian foliation with totally geodesic leaves on $G$ is isometric to a homogeneous foliation.
Luigi Verdiani, Università di Firenze
Prescribed Ricci curvature on cohomogeneity-one manifolds
Given a cohomogeneity-one $G$-manifold we discuss the problem of finding a $G$-invariant Riemannian metric $g$ on $M$ whose Ricci curvature coincides with a prescribed $G$-invariant symmetric $(0,2)$-tensor field $T$ on $M$. Local existence follows, without symmetry assumptions, from the work of DeTurck assuming that $T$ is non degenerate on $M$. In the homogeneous setting the existence is, in general, obstructed. We study the case of a cohomogeneity-one manifold with a singular orbit and we prove a local existence theorem showing that the tangential part of $T$, i.e. the components $T(X,Y)$ for $X,Y$ tangent to the $G$-orbits, can be prescribed. A local existence results for invariant Einstein metrics follows with a similar proof and can be used to show that the result is sharp i.e. the components $T(X,N)$, where $N$ is orthogonal to the $G$-orbits depends, in general, from the tangential components. This is a joint work with W. Ziller.
David Wraith, Maynooth University
Positive Ricci curvature metrics on the sphere: homotopy groups of the observer moduli space
There has been much recent attention on the topology of the space of positive scalar curvature metrics on certain manifolds. This includes the study of moduli spaces of such metrics: the ordinary moduli space formed by quotienting out by the action of the diffeomorphism group, and the observer moduli space, formed by quotienting by the action of the observer diffeomorphism group. This latter group is the group of diffeomorphisms which fix a basepoint, and for which the derivatives act as the identity on the tangent space at the basepoint. The observer moduli space of positive scalar curvature metrics has been particularly fruitful to study, with extensive non-triviality in the homotopy groups detected in many cases. In this talk we focus on the observer moduli space of Ricci positive metrics on spheres, and establish the first non-triviality results for the higher homotopy groups of this space. From a topological point of view, the approach borrows ideas used for positive scalar curvature, however the geometric strategy required is very different. In particular, the geometric techniques are specific to positive Ricci curvature, and do not exploit the structure of the underlying space of positive scalar curvature metrics. This is joint work with Boris Botvinnik and Mark Walsh.
Wolfgang Ziller, University of Pennsylvania
Finsler metrics with constant curvature
In the realm of Finsler metrics there is a surprisingly large class of metrics with constant curvature. Examples are due to Katok-Ziller and in dimension 2 to R.Bryant. We investigate the geodesic flow of such Finsler metrics. This is joint work with R. Bryant, P. Foulon, S. Ivanov and V. Matveev.