I will discuss connected and non-connectedness properties of spaces and moduli spaces of Riemannian metrics with nonnegative sectional and/or positive Ricci curvature on closed and open manifolds, and, in particular, the following result which is joint work with A. Dessai and St. Klaus :

In each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components.

Abstract

Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric operators on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein. In particular, we show that $D$ has constant eigenvalues, and the set of eigenvalues satisfies the restrictions similar to those in the homogeneous case. We give a characterisation of Einstein extension for particular eigenvalue types of $D$, and a complete classification of Einstein extensions of dimension four -- all of them are Einstein solvmanifolds. We study the case when $N$ is a Lie group and under some additional assumptions show that if the extension $M$ is Einstein, it is a solvmanifold.

This is a joint work with D. Alekseevsky

The Calabi-Yau equation is a PDEs system whose study goes back to the celebrated Calabi conjecture. Recently, Donaldson has described how the equation could be generalized in a natural way to the setting of $2$-forms on $4$-manifolds. Donaldson's program, if carried out, would lead to many new and important results in symplectic geometry. Given a $4$-dimensional compact symplectic manifold $(M,\Omega)$ together an $\Omega-$compatible almost-complex structure $J$, the Calabi-Yau equation consists in $$ (\Omega+d\alpha)^2={\rm e}^F\,\Omega^2\,,\quad Jd\alpha=d\alpha $$ where $F\in C^{\infty}(M)$ is given and $\alpha$ is a unknown $1$-form. In contrast to the Kähler case, it is not known if the equation in the almost-complex setting has always a solution.
The talk focuses on the study of the Calabi-Yau equation in torus fibrations, when the initial datum $F$ is invariant by the action of the fiber. It will be showed that in this case the equation reduces to a generalized Monge-Ampère equation on the basis having always a solution.
The last part of the talk will be about some resent advances on the problem in the Kodaira-Thurston manifold. Reference

[1] E.Buzano, A. Fino and L. Vezzoni, The Calabi-Yau equation for $T^2$-bundles over a torus: the non-Lagrangian case, Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), no.3, 281--298.

[2] E. Buzano, A. Fino and L. Vezzoni, The Calabi-Yau equation on the Kodaira-Thurston manifold, viewed as $S^1$-bundle over a $3$-torus, J. Differential Geom. 101 (2015), no. 2, 175--195.

[3] A. Fino, Y.Y. Li, S. Salamon and L. Vezzoni, The Calabi-Yau equation on $4$-manifolds over $2$-tori, Trans. Amer. Math. Soc. 365 (2013), no. 3, 1551--1575.

[4] L. Vezzoni, On the Calabi-Yau equation in the Kodaira-Thurston manifold, preprint.

Bazaikin and Matvienko showed that every closed, simply-connected smooth $4$-manifold with an effective smooth action of the $2$-torus admits an invariant metric with positive Ricci curvature. I will discuss how to extend this result to show that every closed, simply-connected smooth $(n+2)$-manifold with an effective smooth action of an $n$-torus admits an invariant metric of positive Ricci curvature.

The homogeneous affine surfaces have been classified by Opozda (without torsion) and by Arias-Marco and Kowalski (with torsion); they fall into 3 families (Type $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$). The type $\mathcal{A}$ geometries are those where there is a coordinate atlas where the Christoffel symbols are (locally) constant.
We discuss moduli spaces related to these geometries both in the $2$-dimensional and in the higher dimensional settings. This is joint work with M. Brozos-Vazquez , E. García-Río , and J.H. Park .

The Ricci iteration is a discrete analogue of the Ricci flow. Introduced in 2007, it has been studied extensively on Kähler manifolds, providing a new approach to uniformisation.
In the talk, we will define the Ricci iteration on compact homogeneous spaces and discuss a number of existence, convergence and relative compactness results. This is largely based on joint work with Yanir Rubinstein (The University of Maryland).

We compute the conjugate locus with multiplicities of a geodesic that lies in the center of a simply connected, $2$-step nilpotent Lie group $N$ with a left invariant metric. We compare the isometry types of two such nilpotent groups whose conjugate loci for central geodesics are the same in a suitable sense. This comparison is related to work of C. Gordon and E. Wilson in a JDG 1997 paper.

We show that $n$-manifolds with a lower volume bound $v$ and upper diameter bound $D$ whose curvature operator is bounded below by $-\varepsilon(n,v,D)$ also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound nonnegative curvature operator.

We give a new approach to the study of the geometry of Riemannian manifolds with smooth measure that takes a natural torsion free connection as the fundamental object of study. The connection defines notions of Ricci and sectional curvature bounds which are slightly different than ones previously considered. Many obstructions to metrics of positive curvature can be extended to these weighted curvatures. In this talk we'll focus on presenting the following result: A compact locally homogeneous manifold admits a measure with non-negative weighted Ricci curvature if and only if it admits a locally homogeneous metric of non-negative sectional curvature. The talk includes joint works with D. Yeroshkin and L. Kennard .

The spin formalism is a natural setting for treating many Riemannian $G$-manifolds of low dimension, $G$ being a special Lie group, for it allows to extend previous knowledge and provides a nice and uniform picture.
The talk will present joint results with Salamon , and Agricola , Friedrich , Höll , and focus on the cases $G=SU(3)$ and $G_2$.

Let $a(M)$ be the algebraic dimension of a complex manifold $M$ and $h(M)$ be the dimension of its space of holomorphic differentials. We show that, If $M$ is a compact complex nilmanifold, then $a(M) \leq h(M)$. We use it to determine $a(M)$ when $M$ has complex dimension $3$ and also mention a relation with the Kähler rank of $M$.
The talk is based on a joint paper with G. Grantcharov and M. Verbitsky .

We show that any sequence of type III blow downs of an immortal homogeneous Ricci flow solution subconverges to an expanding Ricci soliton. Applications to Einstein solvmanifolds and solvsolitons will be given. This is joint work with Ramiro Lafuente

The Normal Holonomy Theorem, proved by Olmos in [Ol90], asserts that the action of the restricted normal holonomy group on any normal space $\nu_p M$ of a submanifold $M$ of a (real) space form is the holonomy representation of a Riemannian symmetric space. It has many important applications to the study of submanifolds of spaces with constant sectional curvature, specially in the theory of isoparametric submanifolds (see [BCO03]).
The main tool in its proof is the Ricci identity, making it almost impossible to adapt it to more general ambient spaces. In this talk we will summarize some recent results concerning the normal holonomy of submanifolds of complex space forms, comparing and contrasting the properties of the action of the normal holonomy group in real and complex geometry. We shall show that the Normal Holonomy Theorem holds in an important large class of submanifolds, namely the so called $CR$-submanifolds (cf. [DV1]). Finally we will present a Berger-type holonomy theorem which allows us to classify all the complex submanifolds of $\mathbb{C}^n$ or $\mathbb{C}P^n$ with non transitive normal holonomy that generalizes the results presented in [CDO11] (cf. [DV2]).

[BCO03] Berndt, J., Console S. and Olmos, C. Submanifolds and holonomy, 2nd. Ed., Chapman & Hall/CRC , (2016).

[CDO11] Console, S., Di Scala A. J., Olmos C. A Berger type normal holonomy theorem for complex submanifolds, Math. Ann (2011), 351, 187-214.

[DV1] Di Scala, A. J., Vittone F. The normal holonomy of CR-submanifolds. to appear in Osaka J. of Math. (Vol. 54 No.1, 2017)

[DV2] Di Scala, A. J., Vittone F. Mok's characteristic varieties and the normal holonomy group arXiv:1503.01941 (2015)

[Ol90] Olmos, C. The normal holonomy group Proc. Am. Math. Soc. (110) (2004), 813-818.

We study the isometry group of a 2-step nilpotent Lie group equipped with a pseudo-Riemannian metric which is invariant by translations on the left. As in the Riemannian situation the key is to understand the group of isometries fixing the identity element. It was proved that this group could be different to the group of orthogonal automorphisms, even for non-degenerate center. We investigate such group and propose a description in low dimensions.

Let $(M,g)$ be a closed Riemannian manifold of $\dim(M)=n\geq 3$. We say that $u\in C^{\infty}(M)$ is a solution of the Yamabe equation if there exists $\lambda\in \mathbb{R}$ such that $u$ satisfies \begin{equation} a_n\Delta_g u+s_g u=c |u|^{\frac{4}{n-2}}u \label{eq:YE} \end{equation} where $a_n=4(n-1)/(n-2)$ and $s_g$ is the scalar curvature of $(M,g)$.
If $u$ is a positive solution of the Yamabe equation, then $u^{\frac{4}{n-2}}g$ is a Riemannia metric of constant scalar curvature $c$. It is well know that there exists a positive solution of (1) if $\lambda$ and the Yamabe constant of $(M,g)$ have the same sign. A nodal solution of Yamabe equation is a solution of (1) which changes sign. The second Yamabe constant is defined by $$Y^2(M,g):=\inf_{h\in[g]}\lambda_2(L_h)vol(M,h)^{\frac{2}{n}}$$ where $[g]$ is the conformal class of $g$ and $\lambda_2(L_h)$ is the second eigenvalue of the conformal Laplacian of $L_h$. This constant is related with nodal solutions of the Yamabe equation (see [1]).

In this talk we are going to discuss some results about the asymptotic behaviour of the second Yamabe constant of a Riemannian product $(W\times N,g+th)$ $(t>0)$. As a consequence, we are going to prove the existence of nodal solutions for $(W\times N,g+th)$ (provided $t$ large enough). Also, we will discuss the existence of $G-$invariant nodal solutions, where $G$ is a compact subgroup of the isometry group. The equivariant results are part of a work in progress joint with Farid Madani .

References

[1] B. Ammann, E. Humbert, The second Yamabe invariant, Journal of Functional Analysis 235 (2006), 377-412.

[2] G. Henry, Second Yamabe constants on Riemannian products, (2015) arXiv:1505.00981

Let $M$ be a simply connected Riemannian homogeneous space and $N = \Gamma \backslash M$ a Riemannian quotient manifold. The Homogeneity Conjecture says that $N$ is homogeneous if and only if all the elements of $\Gamma$ are isometries of constant displacement. This conjecture is proved when $M$ is Riemannian (or even Finsler) symmetric, in a number of settings where $M$ has enough negative curvature to force bounded isometries to be trivial, and in several settings where $M$ is a Riemannian normal homogeneous space. I'll describe some new results proving the conjecture for a class of homogeneous spaces that includes twistor spaces, and time permitting I'll summarize the current state of the conjecture.

In a 2011 paper, H. Tamaru finds new examples of Einstein solvmanifolds, as submanifolds of noncompact symmetric spaces. His method uses parabolic subalgebras of semisimple Lie algebras to build solvable subalgebras by restricting to a subset $\Lambda'$ of the set $\Lambda$ of simple roots in order to generate a subalgebra. The corresponding submanifold, with the inherited inner product, is called an attached solvmanifold. Tamaru proves that the Ricci curvature is inherited, unchanged. Thus whenever the original solvmanifold is Einstein, the attached subsolvmanifold is also Einstein.
We explore the criteria needed in order for Tamaru's method to be extended to more general Einstein solvmanifolds.
Joint work with T. Payne .

We consider a class of metric 2-step nilpotent Lie algebras associated with simple graphs. First we classify singularity properties of the Lie algebra in terms of the corresponding graph. We then give a necessary and sufficient condition on the graph for the associated Lie algebra to be Heisenberg-like. The density of closed geodesics property has been studied previously for different classes of 2-step nilpotent Lie algebras. In this talk, we will discuss conditions on the graph and on a lattice of the $2$-step nilpotent Lie group associated with the graph for which the nilmanifold has a dense set of smoothly closed geodesics.
This work is joint with Rachelle DeCoste and Lisa DeMeyer.

The eta invariant appears regularly in index theorems but is known to be computable only in certain examples of locally symmetric spaces of compact type.
For general two-step nilmanifolds, the eigenvalues of the Dirac operator are presented as eigenvalues of matrices, with Kirrilov Theory the main too.
This is analogous to the calculation of the Laplace spectrum by H. Pese. For general Heisenberg three-manifolds, the spectrum of the spin Dirac operator and the eta invariant are explicitly computed in terms of the metric, lattice, and spin structure data.
There are continuous families of geometrically, spectrally different Heisenberg three-manifolds whose Dirac operators have constant eta invariant.
We also show that the Dirac spectrum is symmetric about zero in Heisenberg manifolds of dimension $4m+1$, which confirms that the eta invariant is automatically zero in these dimensions.

This work is joint with Ken Richardson (Texas Christian University).

We use evolutionary game theory to analyze the Ricci flow for homogeneous spaces. We start with a brief review of evolutionary game theory. We show that the Ricci flow for homogeneous spaces is the replicator dynamic for a linear or quadratic game. We relate game-theoretic notions to geometric notions; for example, soliton metrics correspond to Nash equilibria. We generalize a well-known formula for computing soliton metrics for homogeneous spaces. We propose using other game dynamics beside the replicator dynamic to define geometric evolution equations. The Ricci flows for several specific classes of homogeneous spaces are analyzed using these methods.

In 1981, M. Gromov completed the proof that every manifold admitting an expanding map is, up to finite cover, homeomorphic to a nilmanifold. Since then it was an open question to give an algebraic characterization of the nilmanifolds admitting an expanding map.
During my talk, I will start by introducing the basic notions of expanding maps and nilmanifolds. Then I explain how the existence of such an expanding map only depends on the covering Lie group and on the existence of certain gradings on the corresponding Lie algebra. One of the applications is the construction of a nilmanifold admitting an Anosov diffeomorphism but no expanding map, which is the first example of this type.

In this talk, we examine the leaf spaces of singular Riemannian foliations and their isometries. In particular, we will review recent results from the literature on smooth isometries of the leaf spaces of singular Riemannian foliations. These leaf spaces include many familiar settings for spectral problems, including manifolds, orbifolds, orbit spaces of G-manifolds, the transverse space of regular Riemannian foliations, as well as more general settings. We also present new results showing that these leaf space isometries preserve the spectrum of the Laplacian on functions on the leaf space, and give insight into the spectral theory of orbifolds, and other leaf spaces of wider interest.

Every quintic Milnor link $K$ on a real $9$-sphere is a nontrivial Sasakian circle fibration over a Calabi-Yau $3$-fold. It admits a canonical co-closed $\mathrm{G}_2$-structure reflecting locally the Calabi-Yau geometry, which induces a natural Yang-Mills theory with many important features of the torsion-free case, such as a Chern-Simons formalism, topological energy bounds and a deformation theory of index $0$. In particular, we obtain $\mathrm{G}_2$-instantons on bundles over $K$ from Hermitian Yang-Mills connections over the quintic Calabi-Yau, thus relating in a concrete example the well-known Donaldson-Thomas invariant with a conjectural $\mathrm{G}_2$-instanton count.

The talk is about degenerations of non-collapsed sequences of Kähler-Einstein metrics on a complex surface with cone singularities along smooth complex curves. This topic has some similarities with the well-stablished theory of Anderson for the case of smooth metrics on $4$-manifolds. The new feature is the behavior of the metrics as the curve degenerates. I will talk about the possible tangent cones on the limit spaces, the blow-up limits and the relations to algebraic geometry.

Simple Lie algebras, their real forms, parabolic subalgebras, and nilradicals of these parabolic subalgebras have been considered by many authors as candidates to find specific geometric structures giving rise to invariant geometric structures on the corresponding Lie groups and their quotients.

The aim of this presentation is to analyze existence of nondegenerate $(0,2)$-tensors on these Lie algebras; in particular invariant symmetric and closed skew-symmetric $(0,2)$-tensors which correspond to bi-invariant metrics and left-invariant symplectic structures at the Lie group level. Among the above candidates, it is well known that semisimple Lie algebras do not admit symplectic structures [Ch] and Borel subalgebras admitting symplectic structures have been described in [AP]. The Killing form on semisimple Lie algebras gives an invariant symmetric nondegenerate bilinear form.

In this talk we focus on the family of nilradicals of parabolic subalgebras. We introduce tools which allow us to give the explicit list of the symplectic nilradicals and those nilradicals admitting invariant metrics. As a first step we find properties which are obstructions for a Lie algebra to admit such tensors. These are cohomological in the skew symmetric case and structural in symmetric one. Afterwards, we prove that almost every nilradical satisfies these obstructions, thus obtaining only few nilradicals admitting invariant tensors. The root system of the simple Lie algebra related to the parabolic subalgebra is fundamental to prove the results.

The presentation is a summary of the results in [CdB] and [dB4].

Reference

[AP] Alekseevsky D. and Perelomov A., Poisson and symplectic structures on Lie algebras. I., J. Geom. Phys., 22 (3) (1997), 191-211.

[CdB] Cagliero L. and del Barco V., Nilradicals of parabolic subalgebras admitting symplectic structures, Differ. Geom. Appl. 46 (2016) 1-13.

[Ch] Chu B., Symplectic homogeneous spaces., Trans. Am. Math. Soc., {\bf 197} (1974), 145--159.

[dB4] del Barco, V., Lie algebras admitting symmetric invariant and nondegenerate bilinear forms preprint arXiv:1602.08286.

This work is joint with Mary Sandoval . We consider the $G$-invariant spectrum of the Laplacian on an orbit space $M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. When such an orbit space is isometric as a metric space to an orbifold we show that the respective $G$-invariant and orbifold spectra are equivalent. We generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric to an orbifold with constant sectional curvature whereas the other admits non-orbifold singularities and therefore has unbounded sectional curvature. We conclude that constant sectional curvature and non-orbifold singularities are inaudible properties of the $G$-invariant spectrum.

The present talk is devoted to the study of the level sets of the polynomials that defines the normal sections on the homogeneous isoparametric hypersurfaces in spheres. We present some new results about homogeneous isoparametric hypersurfaces in spheres, whose number of distinct curvatures is less than or equal four.
Since it is required for our study, we give also explicit expressions for these polynomials.

We classify the Lie algebras of Heisenberg type that are nilradicals of parabolics subalgebras of semisimple Lie algebras, deduce that every simple non-compact Lie Group determines a unique parabolic geometry of that type, and study some of its properties.

To any compact Riemann surface of genus $g >1$ one may assign a principally polarized abelian variety of dimension $g$, the Jacobian of the Riemann surface. The Jacobian is a complex torus and we call a Gram matrix of the lattice of a Jacobian a period Gram matrix. We give explicit estimates for the entries of the period Gram matrix with respect to a suitable homology basis, if the Riemann surface contains a short simple closed geodesic $\gamma$ and study this matrix, if the geodesic is pinched.

We study a half-space problem related to graphs in $\mathbb{H}^2\times\mathbb{R}$, where $\mathbb{H}^2$ is the hyperbolic plane, having constant mean curvature $H$ defined over unbounded domains in $\mathbb{H}^2\times\mathbb{R}$.

In 1994 was formulated the following conjeture, like a possible generalization of the rank rigidity theorem for homogeneous submanifolds: An Irreducible and full homogeneous submanifolds of the sphere, different from a curve, such that the normal holonomy group is not transitive must be an orbit of an s-representation, motevated by that we do a brief introduction to the submanifolds and the relation with the normal holonomy group, and the most recent advances in this geometrical conjeture.

We give the precise asymptotic behavior of residual finiteness for cocompact lattices in connected, simply connected nilpotent Lie groups

We will talk about filiform Lie algebras over fields of characteristic two and we will calculate the cohomology with trivial coefficients of the infinite dimentional Lie algebra $\mathfrak{m}_0$: $[e_1, e_i] = e_{i+1}$ for $2 \leq i$ over $\mathbb{Z}_2$.

We begin by discussing some classical results about geometric properties of manifolds determined by the Laplace and Steklov spectrum. After introducing the concept of an orbifold, we will then look at how these results transfer (or fail to transfer) to the orbifold setting, specifically to the setting of compact $2$-dimensional orbifolds.

In the 70's, Gray and Hervella classified all the almost Hermitian manifolds $\left( M,J,g\right).$ Their work is based on the descomposition in $U(n)$-irreducible components of a subspace $W$ in $TM\otimes TM\otimes TM$ where $\left( \nabla \omega\right)$ belongs. Here, $\omega(x,y)=g(Jx,y)$, is the Kähler form.

We now consider $\left( M,H,g\right)$ as above but now the condition $H^2=-I$ is replaced by the weaker condition: $H$ is skew symmetric. We are interested in manifolds with the extra condition $\nabla H=0$ (parallel condition) or more generally with the weaker condition $(\nabla_x H)x=0 \ \forall \ x\in \ TM $ (Killing-Yano condition). We find examples of these manifolds in Lie groups of dimension $2$, $3$ and $4$ and some examples in flag manifolds.

Let $G$ be a three-dimensional unimodular Lie group, and let $T$ be a left-invariant symmetric $(0, 2)$-tensor field on $G$. We provide the necessary and sufficient conditions on $T$ for the existence of a pair $(g, c)$ consisting of a left-invariant Riemannian metric $g$ and a positive constant $c$ such that $\operatorname{Ric}(g) = cT$, where $\operatorname{Ric}(g)$ is the Ricci curvature of $g$.
We also discuss the uniqueness of such pairs and show that, in almost all cases, there exists at most one positive constant $c$ such that $Ric(g) = cT$ is solvable for some left-invariant Riemannian metric $g$.

I'll provide a brief survey of examples of Ricci solitons and then outline the construction of new steady Ricci solitons on the total space of complex line bundles over a Fano Kähler-Einstein base with first Chern class proportional to that of the base. The construction produces cohomogeneity one solitons when the base manifold is homogeneous. These soliton metrics are typically non-Kahler except in the case of the canonical line bundle.

I'll propose a classical approach to differential supergeometry, in the sense that the sheaf-theoretic approach replaced with concepts closer to classical differential geometry, viz. superalgebra bundles. After proving some results with this perspective I'll explore the group objects in the appropriate categories and show results concerning these Lie supergroups.

In this talk we study left invariant locally conformal symplectic (LCS) and locally conformal Kähler (LCK) structures on solvable Lie groups. Beginning with an LCS (or LCK) Lie algebra and a suitable representation, we give a construction of an LCS (or LCK) structure in the semidirect product of the Lie algebra with the representation space. Finally we study the existence of lattices in the associated simply connected Lie group in order to obtain a solvmanifold equipped with these geometric structures.

We consider the question of which groups arise as isometry groups of solvmanifolds. For those solvable Lie groups which admit an invariant Einstein metric, we show the isometry group of the Einstein metric is maximal in the sense that it contains the isometry group of any other left-invariant metric.
This work is joint with Carolyn Gordon .

The question of when a differentiable manifold admits a Riemannian metric with a particular sign of the curvature is a very natural one. For homogeneous Riemannian manifolds, the case of sectional curvature is quite restrictive and well understood. Concerning negative Ricci curvature, although a great progress has been made lately, specially in the solvable case, the general case seems to be far from being completely understood.
In this talk, we will begin by introducing the known results on negative Ricci curvature in the homogeneous case, and then we will show some unexpected examples of Lie groups with compact Levi factor that admits a metric with negative Ricci curvature.

Topics to be covered in the lectures

• (0) A brief review of Riemannian geometry: Riemannian manifolds, the Levi-Civita connection, different notions of curvature and its geometric and analytic meanings, isometries.
• (1) The Ricci flow: definition, Hamilton’s program, motivations for singularity analysis.
• (2) Ricci solitons: two definitions (dynamical, as self-similar solutions; static, as generalizations of Einstein metrics), historical comments on the name.
• (3) Examples: Einstein manifolds, shrinking, steady and expanding solitons, gradient vs. non-gradient, Gaussian soliton, classification in low dimensions, Kähler Ricci solitons, homogeneous Ricci solitons, the Bryant soliton and other rotationally symmetric examples, solitons from other geometric flows.
• (4) Homogeneous Ricci solitons: shrinkers rigidity, algebraic solitons (definition, examples: nil and solvsolitons), geometric properties, uniqueness, classification.
• (5) Open problems.

References

• [Cao10] Huai-Dong Cao, Recent progress on Ricci solitons, Recent advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 11, Int. Press, Somerville, MA, 2010, pp. 1-38.
• [CCG+07] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey,Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: techniques and applications. Part I - Geometric aspects, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007.
• [Lau09] Jorge Lauret, Einstein solvmanifolds and nilsolitons, New developments in Lie theory and geometry, Contemp. Math., vol. 491, Amer. Math. Soc., 2009, pp. 1–35.
• [Pet06] Peter Petersen, Riemannian geometry, second ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006.
• [Top06] Peter Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series, vol. 325, Cambridge University Press, Cambridge, 2006.

In this course we will focus in some aspects of the scalar curvature on closed Riemannian manifolds. More precisely, let $(M,g)$ be a closed Riemannian manifold. Does there exist a metric, conformal to $g$, with constant scalar curvature? If that is the case, How many of these metrics are? can we prescribe the sign of the scalar curvature? Or even more, Can we prescribe the scalar curvature? Does there exist a Riemannian metric on $M$ with positive scalar curvature? The first question is the celebrated Yamabe problem and its answer, that we know thanks to the works of H. Yamabe [8], N. Trüdinger [7], T. Aubin [1] y R. Schoen [5], is affirmative. The purpose of this course is to give an introduction to curvature scalar problems.

The course is organized as follows. We will begin recalling the definition and some properties of the scalar curvature. We are going to see how the scalar curvature changes within the conformal class of a Riemannian metric. We are going to introduce the Yamabe equation and the Yamabe functional and discuss the relation between the critical points of the latter and positive solutions of the former, as well as the relationship of both with metrics of constant scalar curvature. We will introduce the Yamabe constant, which is an invariant of the conformal class, see examples and properties. In particular we are going to discuss the Yamabe constant of the n-dimensional sphere endowed with the round metric. We are going to treat the subcritical Yamabe equation and its geometric meaning. Afterwards, we will sketch the proof of the Yamabe problem. Finally we will introduce the Yamabe invariant and mention and discuss some result concerning to the sign of the Yamabe invariant. If time permits we will see some results about the problem of prescribing the scalar curvature.

There are course notes in Spanish. However, there is some excellent surveys of these topics in English. To mention some of them: Lee and Parker [4], Aubin [2], Schoen [6], Hebey [3], amongst others.

A short list of references is the following (please, see the complete list in the course notes)

References

• [1] T. Aubin, Équations différentielles non-linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296.
• [2] T. Aubin, Some Non-Linear Problem in Riemannian Geometry, Springer-Verlag, (1998).
• [3] E. Hebey, Variational methods and elliptic equations in Riemannian geometry, Workshop and Conference on Recent Trends in Nonlinear Variational Problems (2003) ICTP, Publications ICTP.
• [4] J. M. Lee, T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17, (1987), 37-91.
• [5] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, Journal of Differential Geometry 20 (1984), 479-495.
• [6] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Lecture Notes in Mathematics 1365, Springer-Verlag, Berlin (1987), 120-154.
• [7] N. S. Trüdinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Annali della Scuola Normale Superiore di Pisa 22 (1968), 265-274.
• [8] H. Yamabe, On a deformation of {R}iemannian structures on compact manifolds, Osaka Mathematical Journal 12 (1960), 21-37.

Contents

• Homologically volume minimizing submanifolds. Examples.
• Alternating multilinear algebra. General facts (review or crash course). Calibrations. Examples.
• The fundamental theorem of calibrations. Review or crash course on differential forms on manifolds, the exterior derivative (with an informal comment on its meaning) and integration of closed forms over homologous submanifolds. Smooth calibrations and the fundamental theorem. Examples.
• Calibrations related to the octonions. The octonions. The associative calibration, examples of volume minimizing cones. The coassociative calibration, examples of volume minimizing graphs with singularities, the Hopf map from $S^3$ to $S^2$.
• Invariant calibrations on homogeneous spaces. The best unit vector field on $S^3$, the Sasaki metric on unit tangent bundles. An application of a $3$-calibration on $\mathbb{R}^6$ with exactly two calibrated subspaces.
• Calibrations in pseudo-Riemannian Geometry.

Bibliography

• F. Reese Harvey, Spinors and calibrations. Perspectives in Mathematics, 9. Academic Press, Boston, 1990.
• John M. Lee, Introduction to smooth manifolds. Graduate Texts in Mathematics, 218. Springer, New York, 2013.

Kähler geometry is a very active and interesting area of current research. It is the aim of this course to provide a quick and modern introduction to different aspects of this subject, with special emphasis on the differential geometric side of Kähler geometry.

The first lecture starts with a short discussion of complex manifolds and holomorphic vector bundles and a detailed account of the basic differential geometric properties of Kähler manifolds.

In the second lecture Kähler manifolds are discussed from the point of view of Riemannian geometry. We give an outline of Hodge and Dolbeault theories, and a simple proof of the famous Kähler identities. The course ends with the study of some aspects of compact Kähler manifolds such as the Calabi conjecture.

This course will be delivered at the VI Workshop on Differential Geometry (EGEO 2016), 1-5 August 2016, to be held in La Falda, Córdoba, Argentina.

References

• [1] W. Ballmann, Lectures on Kähler Manifolds, European Mathematical Society Publishing House, 2006.
• [2] A. Besse, Einstein Manifolds, Chapter 2, Springer-Verlag, 1987.
• [3] X. Chen, Recent progress in Kähler geometry, ICM 2002 Proceedings, Vol. II, 273-282.
• [4] X. Chen, S. Donaldson, S. Sun, On some recent developments in Kähler geometry, 2013,
  http://wwwf.imperial.ac.uk/~skdona/KEDEVELOPMENTS-9-19-2013.PDF
• [5] A. Moroianu, Lectures on Kähler Geometry, London Mathematical Society Student Texts, 2007.

The linear classification of polar representations was later completed by J. Eschenburg and E. Heintze in the irreducible case. It turns out that such representations are determined by the group and a principal isotropy subgroup, though this is no longer true in the reducible case. We show that any polar representation of a connected Lie group is determined by its history and dimension, the former term referring to the collection of isotropy subgroups that appear along a polar section. This is necessary for the reconstruction procedure of K. Grove and W. Ziller.