Finite volume open complete manifolds of negative sectional curvature is largely an uncharted territory, especially when the curvature is not pinched negatively. In the talk I shall survey some results and problems in the field with focus on new examples, collapsing near ends, and relative hyperbolicity of the fundamental group.

Roughly speaking it is expected that the only two types of singular laminations that can occur as limits of closed embedded minimal surfaces in a 3-manifold with positive scalar curvature are accumulations of catenoids and non-properhelicoid-like limits. T. Colding and C. De Lellis constructed an example of the first type. I will present a construction of the second type: we show that there exists a metric with positive scalar curvature on S²×S¹ and a sequence of embedded minimal tori that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. This is a joint work with Darren Lee.

In this talk we discuss a method that succeeds in putting a metric of positive sectional curvature on one of Grove, Wilking and Ziller's candidates for cohomogeneity one 7-manifolds of positive sectional curvature. I will go through the general ideas of the approach and then treat the particular example. I will also discuss other candidates, where one can definitively say the technique as it stands does not work.

In contrast to the positive curvature setting there exist comparatively many examples with non-negative sectional curvature. Hence it is natural to ask whether, among the known examples, it is possible to topologically distinguish manifolds with non-negative curvature from those admitting positive curvature. In joint work with Wolfgang Ziller we address this question. In this talk, after reviewing some of the history, I will describe the topology of two specific families of non-negatively curved manifolds in dimension seven and compare them to known examples of manifolds of positive curvature.

We introduce the concept of genuine isometric / conformal deformation of an Euclidean submanifold and describe the geometric structure of the submanifolds that admit deformations of this kind. That an isometric / conformal deformation is genuine means that the submanifold is not included into a submanifold of larger dimension such that the deformation of the former is given by a deformation of the latter.

Our main results say that an Euclidean submanifold together with a genuine deformation in low, but not necessarily equal codimensions must be mutually (conformally) ruled, and gives a sharp estimate for the dimension of the rulings. Moreover further consequences on the second fundamental form and normal connections are implied.

These results have several strong local and global consequences. Moreover the unifying character and geometric nature of these results, as opposed to a purely algebraic one, suggest that they may be used as the starting point to develop a deformation theory extending the classical Sbrana-Cartan deformation theory for hypersurfaces to higher codimensions.

There are various classical results regarding the topological obstructions to either the existence of metrics of positive scalar curvature or the existence of smooth compact Lie group actions. Such theorems seem to be parallel to each other. Examples of these are the theorems of Lichnerowicz and Atiyah-Hirzebruch on spin manifolds in which the Â-genus figures as the topological obstruction. In this talk we shall build on these parallel stories. LeBrun and Salamon proved vanishing theorems for compact quaternion-Kähler manifolds with positive scalar curvature. We will describe similar vanishing theorems on almost quaternionic manifolds admitting smooth Lie group actions.

Let G/H be a compact, simply conected homogeneous space, where G is a compact, semisimple Lie group and H is closed. The set of G-invariant metrics on M = G/H is determined by the isotropy representation of H on T_pM. We say G/H is isotropy irreducible, if this representation is irreducible, so that the G-invariant metric, which intertwines with the isotropy representation, is unique up to scaling. Since the Ricci curvature tensor must also intertwine with the isotropy representation, it too varies only by a scalar. The unique G-invariant metric is thus automatically an Einstein metric. The (strongly) isotropy irreducible spaces were classified by O. V. Manturov and independently by J. Wolf.

We consider the next-simplest class of homogeneous spaces, where the adjoint action of H on T_pM breaks into exactly two irreducible components. For this class, after fixing the volume, the moduli space of all G-invariant metrics is one-dimensional. We find all such homogeneous spaces M=G/H, where G=SO(n) or G is a simple, compact, connected and simply connected Lie group and H is a connected closed subgroup of G. We describe some interesting results of the classification and some of its uses. This is joint work with Will Dickinson.

La Geometría de los espacios simétricos Riemnnianos es más rica que la de los espacios homogéneos Riemannianos. Sin embargo existe una amplia bibliografía sobre clases especiales de variedades de Riemann homogéneas, las cuales heredan muchas propiedades típicas de los espacios simétricos. Los espacios homogéneos normales, los naturalmente reductivos y los g. o. espacios (que preservan el volumen) son algunos ejemplos interesantes de estas clases de espacios para los cuales, en particular la ecuación de Jacobi puede ser escrita como una ecuación diferencial con coeficientes constantes y su operador de Jacobi tiene rango osculador constante.

Los espacios simétricos compactos de rango uno figuran entre las limitadas variedades conocidas que admiten métricas con curvatura seccional positiva. En efecto existen sólo tres espacios homogéneos normales no simétricos y simplemente conexos con curvatura positiva, V₁=Sp(2)/SU(2) y V₂=SU(5)/Sp(2)×S¹ dados por Berger y V₃=(SU(3)×SO(3))//U(2) descubierta por Wilking. Aquí se demuestran algunas propiedades geométricas de todos estos espacios, las cuales están relacionadas con la existencia de campos de Jacobi isotrópicos. En algunos casos se determina el rango osculador de su operador de Jacobi. También se analizan diferentes subvariedades totalmente geodésicas. Ello da diferentes formas de medir cuanto se desvan estas variedades V_i de los espacios simétricos.

In the first part of the talk we will survey results related to submanifolds and holonomy. We will explain a new application of these methods (joint work with Silvio Reggiani): a Simons type holonomy theorem for 3-forms. We use this result to obtain the following theorem: The full isometry group of a naturally reductive space, which is locally irreducible and not a rank one symmetric space, coincides with the affine transformations with respect to the canonical connection, which can be easily computed.

Given a metric g one looks for unit volume metrics of constant scalar curvature in its conformal class. The Yamabe constant Y(M,[g]) of the conformal class of g is the minimum of such scalar curvatures. Good estimates on Y(M,[g]) are hard to find when the scalar curvature is positive. We will discuss results on finding such estimates in terms of the isoperimetric profile of g.

A G-structure on a differentiable manifold M is a G-principal sub-bundle of the frame bundle of TM. Given a connection on a manifold with G-structure, the inner torsion of the structure is a tensor that measures the lack of compatibility between the connection and the G-structure. An affine manifold with G-structure is said to be infinitesimally homogeneous if the curvature and the torsion of the connection, as well as the inner torsion of the G-structure, are constant in the frames of the structure. The central result of my talk will be an existence theorem of affine immersions that preserve the G-structure, having prescribed second fundamental form and normal connection, between infinitesimally homogeneous affine manifolds with G-structure. I will discuss a few applications in the context of isometric immersions.

In this talk we will discuss some variational problems inside the sub-Riemannian Heisenberg group H¹. The existence in this space of natural notions of length, volume and area lead us to study sub-Riemannian analogs of Riemannian objects like geodesics and constant mean curvature sufaces. By using techniques of Riemannian geometry we shall describe in more detail recent results about area-minimizing surfaces with or without a volume constraint in H¹.

Let X be a CAT(-1) space and o a basepoint in X. Bourdon showed that the boundary at infinity of X carries a natural metric d_o. We prove that this metric satisfies the Ptolemy inequality and give some applications of this result.

Nearly Kähler manifolds are special almost hermitian manifolds. In dimension 6 they have many interesting properties, e.g. the metric is Einstein, the canonical hermitian connection is skew symmetric with parallel torsion and there exist Killing spinors. However there are only very few known examples.

In my talk I will introduce the class of nearly Kähler manifolds, with its basic properties, examples and some classification results. Then I will discuss the question whether it is possible to deform such manifolds. It turns out that the space of infinitesimal deformations can be identified with a certain eigenspace of the Laplace operator on 2-forms.

The talk will begin with a survey of known results concerning the topology of complete noncompact manifolds with nonnegative Ricci curvature and additional conditions on volume growth, diameter growth or dimension. Geometrically intuitive descriptions of the proofs of key results will be given. Recent results by Michael Munn will be described as well. The talk will end with open questions and a description of a possible counter example to Milnor's 1968 Conjecture using a dyadic solenoid complement. Actual implementation of such an example would require a new kind of twisted warped product that could perhaps be implemented in a joint effort with other participants.

I will begin with the question: Classify the left-invariant metrics with nonnegative sectional curvature on a given compact Lie group. I will then discuss applications of the natural power-series approach to this problem, including a partial classification of cohomogeneity-one manifolds with nonnegative curvature and at least one totally geodesic principle orbit and new examples of homogeneous metrics with nonnegative curvature.

We will talk about methods for constructing metrics of positive sectional curvature on a class of cohomogeneity one manifolds in dimension 7. We will present numerical evidence that at least in one case it is indeed possible to construct such an example. We will also outline a strategy for a rigorous proof of this result. This is joint work with W. Ziller and K. Grove.

We call a metric quasi-Einstein, if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and get several rigidity results. We also give a splitting theorem for some Kähler quasi-Einstein metrics. This is a joint work with J. Case and Y. Shu.

In the classification of positively curved cohomogeneity one manifolds, a series of candidates for positive curvature arose in dimension 7. We describe the geometry of these manifolds and their surprising connection to self dual Einstein metrics.