Camillo Brena (Scuola Normale Superiore, Pisa)
Title: Unique continuation for area minimizing currents
Abstract: Consider an m-dimensional area minimizing current and an m-dimensional minimal surface. If, in an integral sense, the current has infinite order of contact with the minimal surface at a point, then the current and the minimal surface coincide in a neighbourhood of that point.
These results are contained in a work with Stefano Decio.
Albachiara Cogo (Universtät Tübingen)
Title: Sobolev conformal structures on closed 3-manifolds
Abstract: Riemannian structures of limited regularity arise naturally in the realm of geometric PDEs. It is well known, since the work of Sabitov-Shefel and De Turck-Kazdan in the late seventies, that the optimal regularity of a Riemannian structure is governed by that of the Ricci tensor in harmonic coordinates.
In this talk, we will discuss the conformal analogue problem. More precisely, we consider a
closed three-dimensional Riemannian manifold (M, g) with g in the Sobolev class W2,q with q > 3 and show that the optimal regularity of the conformal structure is governed by that of the conformally invariant Cotton tensor. The proof requires the resolution of the Yamabe problem for W2,q metrics, which is also interesting on its own.
This is based on a joint work with R. Avalos and A. Royo Abrego.
Joshua Daniels-Holgate (Hebrew University of Jerusalem)
Title: Mean curvature flow from conical singularities
Abstract: We discuss some regularity results for mean curvature flow from smooth hypersurfaces with conical singularities.
We then discuss how to use these results to tackle two conjectures of Ilmanen.
Luigi De Masi (Università degli studi di Trento)
Title: Regularity of capillary minimal surfaces
Abstract: In this talk I will introduce an Allard-type regularity result for minimal hypersurfaces in a Riemannian manifold which meet the boundary with a prescribed contact angle. More precisely, if a surface is stationary for a capillary functional and is sufficiently close in a very weak sense to a half-plane, then it is a C1,α graph over that plane with uniform estimates. The main ingredients are a "first variation control" (a fact with its own interest) and viscosity techniques. This is a joint work with N. Edelen, C. Gasparetto and C. Li.
Matilde Gianocca (ETH, Zürich)
Title: Morse Index Stability for Harmonic Maps and their Min Max Construction
Abstract: I will explain the behaviour and compactness properties of sequences of (approximate) harmonic maps in dimension two. The well-known energy identity shows that the energy in the limit is equal to the energy of the weak limit plus the energy of finitely many bubbles, all of which are harmonic maps. In collaboration with T. Rivière and F. Da Lio we proved that the extended Morse Index is upper semi-continuous, by improving the neck region analysis underlying the energy identity.
The main steps of the proof will be explained and, if time permits, I will discuss how to extend the result to Ginzburg-Landau sequences approximating harmonic maps into the sphere (in collaboration with F. Da Lio)
Florian Johne (Albert-Ludwigs-Universität Freiburg)
Title: Positive intermediate curvature and differential forms
Abstract: Positive intermediate curvature is a notion of curvature interpolating between Ricci curvature and scalar curvature. The use of (weighted) stable minimal hypersurfaces allows to prove non-existence results for positive intermediate curvature on manifolds
of topology Nn =𝕋m x𝕊m. In this talk, we will explore how to study positive intermediate curvature with the help of differential forms.
Andrea Merlo (Universidad del País Vasco)
Title: Free Boundary Problems for elliptic measure for elliptic operators satisfying the Dahlberg-Kenig-Pipher conditions
Abstract: TBA
Andrea Nützi (Stanford University)
Title: A support preserving homotopy for the de Rham complex with boundary decay estimates
Abstract: I will explain the construction of a chain homotopy for the de Rham complex of relative differential forms on compact manifolds with boundary. This chain homotopy has desirable support propagation properties, and satisfies estimates relative to weighted Sobolev norms, where the weights measure decay near the boundary. The estimates are optimal given the homogeneity properties of the de Rham differential under boundary dilation, and are obtained by showing that the homotopy is a b-pseudodifferential operator. When applied to the radial compactification of Euclidean space, this construction yields, in particular, a right inverse of the divergence operator that preserves support on large balls around the origin, and satisfies estimates that measure decay near infinity. We will compare this to a right inverse obtained previously by Bogovskii, which also preserves support, but whose mapping properties are not optimal with respect to decay.
Lauro Silini (ISTA)
Title: Characterizing isoperimetric regions in weighted manifolds
Abstract: In this talk, after an introduction to the isoperimetric problem in general weighted manifolds, we will discuss the state of the art in the Euclidean and hyperbolic setting under the assumption of radial symmetry of the weight. In particular, we will show how the classification of isoperimetric regions in weighted hyperbolic planes implies the optimality of geodesic spheres in (unweighted) symmetric manifolds under the assumption of suitable central symmetry of the competitors.