Titles and Abstracts

Luciano Mari: On the 1/H flow via p-Laplace approximation under Ricci lower bounds.

Abstract: This talk is about the inverse mean curvature flow (IMCF) on complete Riemannian manifolds. In the first part, we will introduce the flow on asymptotically flat manifolds, and sketch its use to prove the celebrated Penrose inequality in General Relativity. After discussing the necessity of a weak notion of IMCF, we will study the existence problem for the IMCF on a complete manifold with only Ricci lower bounds. We will consider both solutions issuing from a point and from the boundary of a relatively compact open set. To prove their existence (in the sense of Huisken-Ilmanen), we follow the strategy pioneered by R. Moser using approximation by p-Laplacian kernels. In particular, we prove new and sharp gradient estimates for the kernel of the p-Laplacian on M via the study of the fake distance associated to it. We address the compactness of the flowing hypersurfaces, and time permitting some monotonicity formulas that relate to recent works of Colding-Minicozzi and Agostiniani-Mazzieri-Pinamonti Borghini-Fogagnolo. This is based on joint work with M. Rigoli and A.G. Setti.

Roberto Paoletti: Equivariant asymptotics for Szego kernels.

Abstract: Let M be a complex projective manifold, and let A be a positive line bundle on M; M is then endowed with a naturally associated Kahler structure. Suppose that a connected and compact Lie group G acts on M in a holomorphic and Hamiltonian manner, and that this action can be linearized to A. Then there is an intrisically induced unitary representation of G on the Hardy space associated to the polarization. In the classical case of the trivial action of S^1 on M, with moment map \Phi =1, the isotypical components are, essentially, the spaces of global holomorphic sections of the tensor powers of A. More generally, whenever the moment map is nowhere vanishing the isotypical components are all finite-dimensional, but they do not necessarily correspond to spaces of sections of some power of A. It is natural to investigate the local and global asymptotic properties of the corresponding equivariant Szego projectors associated to a weight of the form $k \nu$, when $k\rightarrow +\infty$. We shall discuss some results in the literature, and then describe some recent progress in the cases G=SU(2) and G=U(2) (joint work with A. Galasso).

Alberto Raffero: Closed G_2-structures with symmetry.

Abstract: G_2-structures defined by a closed positive 3-form constitute the starting point in various known and potentially effective methods to obtain holonomy G_2 metrics on seven-dimensional manifolds. Currently, no general results guaranteeing the existence of such structures on compact 7-manifolds are known. Moreover, the construction of new explicit examples requires substantial efforts.

In the introductory part of this talk, aimed at undergraduate students, I will explain the basics of G_2-geometry.

In the research part, I will first discuss the properties of the automorphism group of a compact 7-manifold endowed with a closed G_2-structure, showing how they impose strong constraints on the construction of examples with high degree of symmetry (e.g. homogeneous, cohomogeneity one). Then, I will focus on the non-compact case, discussing the existence of left-invariant closed G_2-structures on certain types of unimodular Lie groups.

Marco Rigoli: Einstein-type structures.

Abstract: Given a Riemannian manifold M we introduce a modification of the curvatures and of their covariant derivatives to relate the geometry of M to that of a smooth map from M to a second Riemannian manifold N. This enables us to study a general structure, that we call an Einstein-type structure, that contains and extends various special cases: notably Ricci solitons, generalised quasi-Einstein manifolds, Ricci-harmonic solitons and so on. We prove rigidity/uniqueness results in the compact and in the complete case. We also characterize the local geometry in a non degenerate setting that we shall justify geometrically.