Research Talks

Ian Adelstein (Yale)
A characterization of round spheres in terms of half-geodesics

Half-geodesics are those closed geodesics $c: [0, 2\pi] \to M$ that minimize between every pair of points $c(t)$ and $c(t + \pi)$. The great circles on the round sphere are examples of half-geodesics. We'll first provide a relationship with the Grove-Shiohama notion of critical points of distance to prove some results on the behavior of half-geodesics. A characterization of round spheres in terms of half-geodesics will be given using recent results by Radeschi-Wilking and Lin-Schmidt.

Stephanie Alexander (UIUC)
Spacetime analogues of Riemannian comparisons in curvature bounded below

We consider spacetime concave  functions and global comparisons in Lorentzian manifolds, assuming either (i) timelike sectional curvatures $\le K$; or (ii) timelike sectional curvatures $\le K$ and spacelike sectional curvatures $\ge K$. (Condition (ii) is the semi-Riemannian analogue of Riemannian sectional curvature $\ge K$.)

Thomas Brooks (UPenn)
3-Manifolds with Constant Ricci Eigenvalues λ, λ, 0

We consider the complete Riemannian 3-manifolds whose Ricci tensor has constant eigenvalues λ, λ, 0 and obtain a classification of those satisfying a regularity condition. Such manifolds are the curvature homogeneous examples of constant vector curvature 0 (cvc(0)). Moreover, they are all the curvature homogenoeus manifolds whose nullity of curvature tensor is one. By a result of Szabo, the classification of curvature homogeneous semi-symmetric 3-manifolds reduces to considering these manifolds.

Luis Florit (IMPA)
Manifolds with conullity at most two as graph manifolds

We find necessary and sufficient conditions for a complete Riemannian manifold of finite volume whose curvature tensor has conullity at most 2 to be a geometric graph manifold. In the process, we show that Nomizu's conjecture, well known to be false in general, is true for manifolds with finite volume.

Carolyn Gordon (Dartmouth)
Symmetry properties of homogeneous Einstein and Ricci soliton metrics

We say that a left-invariant Riemannian metric on a Lie group is maximally symmetric  if, up to automorphisms, its isometry group contains that of any other left-invariant metric. We also define a weaker notion of infinitesimal maximal symmetry. We find that Einstein metrics on solvable Lie groups are maximally symmetric, while expanding Ricci solitons on solvable Lie groups of real type are infinitesimally maximally symmetric. We give counterexamples showing that such Ricci solitons need not be maximally symmetric. We also relate these results to questions of stability of the Ricci flow. This is joint work with Michael Jablonski.

Anusha Krishnan (UPenn)
Cohomogeneity one Ricci flow and non negative curvature

We show that $\mathbb{S}^4, \mathbb{C}P^2, \mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{C}P^2 \# -\mathbb{C}P^2$ admit metrics of nonnegative sectional curvature which immediately lose this property under the Ricci flow. Although this was previously known for compact manifolds of dimension $> 5$ and for non-compact manifolds, these are the first compact 4-dimensional examples showing such behaviour, and show some limitations of the Ricci flow above dimension 3 (where nonnegative sectional curvature is preserved). Our approach involves studying the Ricci flow on manifolds admitting an isometric cohomogeneity one group action. This talk is based on joint work with Renato Bettiol.

Heather Macbeth (MIT)
Kahler-Ricci solitons on crepant resolutions

By a gluing construction, we produce steady Kahler-Ricci solitons on crepant resolutions of $\mathbb C^n/G$, where $G$ is a finite subgroup of $SU(n)$, generalizing Cao's construction of such a soliton on a resolution of $\mathbb C^n/\mathbb Z_n$. This is joint work with Olivier Biquard.

Marco Radeschi (Notre Dame)
Minimal Hypersurfaces in compact symmetric spaces

A conjecture of Marquez-Neves-Schoen says that for every closed embedded minimal hypersurface M in a manifold of positive Ricci curvature, the first Betti number of M is bounded above linearly by the index of M. We will show that for every compact symmetric space this result holds, up to replacing the index of M with its extended index. Moreover, for special symmetric spaces, the actual conjecture holds for all metrics in a neighbourhood of the canonical one. These results are a joint work with R. Mendes.

Catherine Searle (Wichita State)
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
(1)  $k\leq \lfloor 2n/3\rfloor$;
(2)  When $k= \lfloor 2n/3\rfloor$, $M^n$ is equivariantly diffeomorphic to $$Z= \prod_{i\leq r} \mathbb{S}^{2n_i+1} \times\prod_{i>r} \mathbb{S}^{2n_i}, \, \, \, \, {\textit{ with }} r= 2\lfloor 2n/3\rfloor-n,$$ or the quotient of $Z$ by a free linear action of a torus of rank less than or equal to $2n$ mod $3$.

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 (MPI 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.

Burkhard Wilking (Münster)
The topology of fixed point components in positive curvature

We consider isometric effective 5-torus actions on positively curved manifolds. We show that each fixed point component has either the rational homotopy type of a rank 1 symmetric space or of $\mathbb S^k \times \mathbb HP^l$ ($k=2,3$). This is joint work with Lee Kennard.