Talks (titles and abstracts)
Ricardo Caniato: Variations of the Yang-Mills Lagrangian in high dimension
In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under \(L^2\)-control of their curvature. We will start from the classical theory in critical dimension and then move to the super-critical dimension, making a digression about the so called “abelian” case and thus showing an analogy between p-Yang-Mills fields on abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to build a Schoen-Uhlenbeck type regularity theory for Yang-Mills fields in supercritical dimension.
Guido De Philippis: Decay of excess for the abelian Higgs model
Entire critical points of the Yang–Mills–Higgs functional are known to blow down to (generalized) minimal surfaces. Goal of the talk is to prove an Allard's type large scale regularity result for the zero set of the solution. In particular, in the "multiplicity one" energy regime, we show uniqueness blow-downs and we classify entire solutions in small dimensions and of entire minimizers in any dimension.
This is based on a joint work with Aria Halavati and Alessandro Pigati.
Giada Franz: Construction and properties of free boundary minimal surfaces via min-max
A free boundary minimal surface (FBMS) in a three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold. It is natural to ask about the existence of FBMS (in a given ambient manifold) and their properties (topology, area, Morse index, etc.).
In this talk, we will analyze these questions through the lens of Simon-Smith variant of Almgren-Pitts min-max theory. More precisely, we will see how this method allows the construction of FBMS with prescribed properties (symmetry, topology, Morse index, etc.), by presenting new developments and discussing the limits and perspectives of this approach.
Ailana Fraser: Higher codimension minimal surfaces in Riemannian geometry
A fundamental question in Riemannian geometry is to understand the relationships between the curvature and the geometry and topology of Riemannian manifolds. The classical theorem of Bonnet-Myers gives an upper bound on the length of any stable geodesic in terms of a lower positive bound on the Ricci curvature. In this talk we will discuss Bonnet-Myers type theorems for stable minimal surfaces in manifolds with positive isotropic curvature.
Lan-Hsuan Huang: Structure theory of Einstein manifolds with boundary
We discuss results on the structure of compact Einstein manifolds in terms of the conformal boundary metric and boundary mean curvature. In three dimensions, we confirm M. Anderson's conjecture, showing that the map from the space of Einstein metrics to such boundary data is generically a local diffeomorphism. Additionally, we discuss analogous results in dimensions greater than three for Ricci flat manifolds or Einstein manifolds with a negative constant, assuming the non-degenerate boundary condition. This talk is based on joint work with Zhongshan An (University of Michigan).
Paul Laurain: Stability of the Yang-Mills Connections Index
After introducing the setting of Yang-Mills connection, I will review some classical works on Yang-Mills in critical dimension (dimension 4), such as the gauge fixing by Uhlenbeck and the energy quantization. Then, I will demonstrate how, with Mr. Gauvrit, we have proved the semi-continuity for the index of sequences of Yang-Mills connections. Finally, I will give some perspectives on the construction of non-selfdual connections.
Yangyang Li: Existence and regularity of anisotropic minimal hypersurfaces
Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. The regularity theory for its critical points, anisotropic minimal hypersurfaces, is significantly more challenging than the area functional case, mainly due to the absence of a monotonicity formula. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface and optimally regular minimal hypersurfaces for elliptic integrands in closed Riemannian manifolds through min-max construction. This confirms a conjecture by Allard in 1983. If time permitted, I will also discuss how this could be connected to the minimal surface theory. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.
Andrea Malchiodi: Yamabe metrics on conical manifolds
We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin’s classical result. The singular nature of the metric determines a different condition on the dimension, compared to the regular case.
We derive asymptotic expansions on the Yamabe quotient by adding a proper and implicit lower-order correction to standard bubbles, whose contribution to the expansion of the quotient can be determined combining the decomposition of symmetric 2-tensor fields and Fourier analysis on the conical links. This is joint work with M. Freguglia.
Christos Mantoulidis: Improved generic regularity for minimizing hypersurfaces
I will discuss recent and ongoing work with O. Chodosh, F. Schulze, and Z. Wang showing that minimizing hypersurfaces have, after a suitable perturbation, a smaller than codimension-7 singular set.
Lorenzo Mazzieri: On the positive mass problem for initial data with a positive cosmological constant
The concept of mass for time-symmetric initial data has been extensively explored and is now a cornerstone in the study of contemporary Mathematical General Relativity, especially in relation to spacetimes with zero or negative cosmological constants. However, the case of a positive cosmological constant presents a distinct challenge, as our understanding is still unsatisfactory at the present stage. The renowned counterexample by Brendle, Marques, and Neves to the Min-Oo conjecture highlights that even the rigidity statement in a potential positive mass theorem has not been correctly identified yet in this context. In this presentation, I will propose approaches to address this issue and, if time allows, explore applications in characterizing the de-Sitter spacetime.
Alexis Michelat: Geometric Aspects of Willmore Immersions
In this talk, we will review some recent and classical results on the more geometric aspects of Willmore surfaces—immersions that are the critical points of a conformally invariant functional. Going beyond the pure minimisation of the Willmore energy, we will describe a general class of natural min-max problems it can be applied to, and explain what can be said about the Morse index of Willmore immersions and its stability under weak convergence.
Paul Minter: Fine structure of singularities in area minimising currents mod(q)
One setting that one may study the Plateau problem is in the class of area minimising currents modulo a positive integer \(q\). The regularity questions which arise are closely related to those for area minimisers, but fundamentally different due to the possibility of codimension one singularities which are not present for area minimisers. I will discuss recent work (joint with Camillo De Lellis and Anna Skorobogatova) which provides a rather satisfactory description of the singular set of such an \(m\)-dimensional area minimiser mod \(q\), showing that, up to a countably \((m-2)\)-rectifiable set, it is a \(C^{1,\alpha}\) manifold of dimension \(m-1\). We are also able to establish uniqueness of tangent cones for various subsets of the singular set.
André Neves: Minimal surfaces in negatively curved manifolds
I will talk about my recent work with Fernando Marques where among other things we find sharp lower bounds for the product of the area of an essential minimal surface and a closed geodesic, if they are chosen randomly.
Alessandro Pigati: Topology of three-dimensional Ricci limits and RCD spaces
In the class of n-dimensional complete Riemannian manifolds, it was observed by Gromov that a lower bound on the Ricci curvature is the essential ingredient in order to control the number of degrees of freedom at a metric level, allowing to compactify (in a very weak sense) the subclass of manifolds obeying a Ricci lower bound. It was later understood that regular or singular spaces belonging to this compactification (Ricci limits) are special cases of a more general analytic notion (RCD spaces), which is more stable with respect to certain natural operations, and thus they also inherit a rich analytic structure, allowing to do calculus on them.
In this talk, based on joint work with Elia Bruè and Daniele Semola, we will review some previously known structural results for Ricci limits and RCDs in the non-collapsed case, as well as some instructive examples and counterexamples, and we will see a new, more elementary proof that Ricci limits of dimension three are generalized manifolds, enjoying in particular uniform contractibility. Our tools, together with some results in geometric topology, give an alternative proof that they are in fact topological manifolds, which was first shown by Simon-Topping using deep results on the Ricci flow. We will also see a new result for tangent cones in higher dimension; the latter is based on a new topological regularity and stability theorem for RCDs in dimension three.
Antonio Ros: Minimal surfaces and the first eigenvalue of the Laplacian
For a given genus, the first normalized eigenvalue of the Laplacian is a natural functional defined on the space of closed Riemannian surfaces. Critical points, in particular the maxima, and their relationships with conformal geometry and minimal surfaces of the sphere is an interesting and active field of research. We will consider these issues by focusing on the small genus case.
We present a new proof of the theorem of Nayatani and Shoda which states that the first eigenvalue of the spherical Bolza surface is equal to 2 and, as a consequence, gives that for genus=2 the maximum of the functional is \(16\pi\). For genus=3 we construct a minimal immersion of the Klein quartic in the 7-sphere by first eigenfunctions and it is reasonable to conjecture that this minimal surface provides the maximum of the first eigenvalue for genus=3.
Thomas Schick: Rigidity of positive scalar curvature under low regularity
In this talk we discuss extremality and rigidity properties of positive scalar curvature metrics. This builds on conjectures of Gromov and on work of Llarull (and others).
Main result: A map f: X -> S^n from a compact Riemannian spin manifold (X,g) to the standard sphere which is 1-Lipschitz and of non-zero degree and such that the scalar curvature of X is everywhere greater or equal to the one of the sphere is automatically an isometry. Here, it is sufficient that the metric is only a Lipschitz metric (which distributional scalar curvature) and that the map is only Lipschitz continuous, not necessarily smooth. New ingredients of proof: combination of Dirac operator tools and the theory of quasiconformal maps. Low regularity index theory on manifolds with conical singularities.
Coauthors: Simone Cecchini, Bernhard Hanke, recently also Lukas Schönlinner
Richard Schoen: Stable minimal surfaces in higher codimension
In this lecture we will discuss geometric and analytic properties of stable minimal surfaces in higher codimension. We will focus mainly on the tendency for high curvature regions to be nearly holomorphic. This is reflected in Bernstein-type theorems for complete stable surfaces in euclidean space. We will summarize known results and recent additions to the topic.
Felix Schulze: Mean Curvature Flow from conical singularities
We give a proof of Ilmanen's resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. Combined with the uniqueness of asymptotically conical tangent flows, this shows how the outermost mean curvature flows evolve through such singularities and how mean curvature flow becomes non-unique past such singularities. Furthermore, we resolve a particular case of Ilmanen's strict genus reduction conjecture.
Precisely, we prove that the level set flow of a smooth hypersurface \(M^n \subset R^{n+1}\), \(2\leq n \leq 6\), with an isolated conical singularity is modelled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens. This is joint work with Otis Chodosh and Joshua Daniels-Holgate.
Carlo Sinestrari: Mean curvature flow in asymptotically flat three manifolds in General Relativity
I will talk about the volume preserving mean curvature flow in an asymptotically flat three manifold modeling an initial data set in General Relativity with positive mass. I will present an extension to a larger class of ambient manifolds of a well-known result by Huisken-Yau, who proved that there is a class of round surfaces for which the flow exists globally and converges to a CMC limit. In contrast to the Huisken-Yau setting, in our case one the geometry of the evolving surfaces can no longer be controlled by maximum principle arguments and one has to use integral estimates instead. This provides an alternative approach to the construction of CMC foliations in asymptotically flat spaces obtained by various authors through the years with different methods not involving flows, e.g. Huang, Metzger and Nerz. A similar analysis also applies for the more general flow with speed given by the space-time mean curvature recently considered by Cederbaum-Sakovich. These results are in collaboration with J. Tenan (Roma "Tor Vergata").
I will also discuss another foliation construction of asymtptotically flat spaces, obtained instead by the standard (non volume-preserving) mean curvature flow. One can show that there exists a unique ancient (i.e., defined for all negative times) solution of the flow, foliating the end of the manifold by surfaces which are asymptotically round and CMC. The positivity of the mass is again a crucial hypothesis for the stability of the flow and to berak the translation invariance of the Euclidean case. These results are in collaboration with G. Huisken (Tübingen).
Daniel Stern: New minimal surfaces in \(S^3\) and \(B^3\) via eigenvalue optimization
I'll describe joint work with Karpukhin, Kusner, and McGrath, in which we produce many new families of closed minimal surfaces in \(S^3\) and free boundary minimal surfaces in \(B^3\) via constrained optimization problems for Laplace and Steklov eigenvalues on surfaces. Along the way, I'll highlight some new techniques for establishing existence of extremal metrics in more general situations, and point to some stubborn open problems.
Michael Struwe: The prescribed curvature flow on the disc
For given functions \(f\) and \(j\) on the disc \(B\) and its boundary \(\partial B=S^1\), we study the existence of conformal metrics \(g=e^{2u}g_{\R^2}\) with prescribed Gauss curvature \(K_g=f\) and boundary geodesic curvature \(k_g=j\). Using the variational characterization of such metrics obtained by Cruz-Blazquez and Ruiz in 2018, we show that there is a canonical negative gradient flow of such metrics, either converging to a solution of the prescribed curvature problem, or blowing up to a spherical cap. In the latter case, similar to our work in 2005 on the prescribed curvature problem on the sphere, we are able to exhibit a \(2\)-dimensional shadow flow for the center of mass of the evolving metrics from which we obtain existence results complementing the results recently obtained by Ruiz by degree-theory.
Douglar Stryker: Stable minimal hypersurfaces in R^5
I will discuss why every complete two-sided stable minimal hypersurface in R^5 is flat, based on joint work with Otis Chodosh, Chao Li, and Paul Minter.
Yipeng Wang: Rigidity Results in Scalar Curvature
A central theme in Gromov's program is the exploration of the implications of metric geometry for spaces with scalar curvature bounded from below. Inspired by Toponogov's triangle comparison theorem on manifolds with nonnegative sectional curvature, Gromov postulated a conjecture regarding the scalar curvature extremality property of convex polytopes. In his 'Four Lectures', he also outlined a proof based on a smoothing procedure. In this talk, I will discuss some recent progress on this problem, as well as several aspects of its generalizations and open questions.
Brian White: Translating Annuli for Mean Curvature Flow
I will discuss suprising new examples of complete minimal annuli that translate under mean curvature flow.
Zhizhang Xie: On Gromov’s dihedral extremality/rigidity conjecture of scalar curvature
In this talk, I will present my joint work with Jinmin Wang and Guoliang Yu on a new index theorem for manifolds with singularities (such as manifolds with corners and more generally for manifolds with polyhedral type boundary). As an application, we obtained a positive solution to Gromov’s dihedral extremality/rigidity conjecture. This conjecture concerns comparisons of scalar curvature, mean curvature and dihedral angles for compact manifolds with polyhedral type boundary, and has very interesting implications in geometry and mathematical physics. Further developments of this new index theorem have led us to a positive solution of Gromov's flat corner domination conjecture. As a consequence, we answered positively a long standing conjecture in discrete geometry - the Stoker conjecture.