Abstracts
Robert Bryant
On the Affine Bonnet Problem
The Bonnet problem in Euclidean surface theory is well-known: Given a metric g on an oriented surface M and a function H, when (and in how many ways) can (M,g) be isometrically immersed in Euclidean 3-space with mean curvature H? For generic data (g,H), such an isometric immersion is impossible and, in the case that it does exist, the immersion is unique. Bonnet showed that, aside from the famous case of surfaces of constant mean curvature, there is a finite dimensional moduli space of (g,H) for which the space of such Bonnet immersions has positive dimension.
The corresponding problem in affine theory (a favorite topic of Eugenio Calabi) is still not completely solved. After reviewing the Euclidean results on this problem by O. Bonnet, J. Radon, É. Cartan, and A. Bobenko, I will give a report on affine analogs of these results. In particular, I will consider the classification of the data (g,H) for which the space of affine Bonnet immersions has positive dimension, showing a surprising connection with integrable systems in the case of data with the highest possible dimension of solutions.
David Calderbank
Differential geometric ideas in geometric analysis: a case study in Sasaki geometry
Sasaki geometry is a geometry in odd dimensions closely related to Kahler geometry in even dimensions, and there is considerable interest in finding distinguished metrics along the same lines as the recent proof of the existence of Kahler-Einstein metrics on K-stable Fano manifolds. A more general class of distinguished Kahler metrics are the extremal metrics of Calabi, and they have an analogue in Sasaki geometry, where they are essentially metrics which are transversal extremal Kahler metrics.
The equation for extremal Sasaki metrics is a fourth order nonlinear PDE, but its geometric origins make it approachable. In joint work with Vestislav Apostolov and Eveline Legendre we made progress on existence and constructions of extremal Sasaki metrics. The results are not the current state-of-the-art, so the work is more interesting for the differential geometric ideas that arose naturally, in particular the role of CR invariance. Hence this talk will focus on these ideas and methods in the hope that similar techniques could have a wider currency in other geometric analysis problems.
Isabel Fernández
Free Boundary Minimal & CMC Annuli in the Ball
We will present a family of free boundary minimal annuli immersed in the unit ball of Euclidean 3-space, the first such examples other than the critical catenoid. We will also construct embedded free boundary CMC annuli and embedded capillary minimal annuli in the unit ball that are not rotational.
In the minimal case, this construction answers in the negative a problem of the theory that dates back to Nitsche in 1985, who claimed that such annuli could not exist. In the CMC case, the existence of these embedded annuli is seemingly unexpected, and solves a problem by Wente (1995). Joint work with Mira-Hauswirth (for the minimal case) and Cerezo-Mira (for the CMC case).
Panagiotis Gianniotis
Splitting maps in Type I Ricci flows
Harmonic almost splitting maps are an indispensable tool in the study of the singularity structure of non-collapsed Ricci limit spaces. In fact, by recent work of Cheeger-Jiang-Naber the singular stratification is rectifiable, and almost splitting maps can be used to construct bi-Lipschitz charts of the singular strata. For this, it is crucial to understand how a splitting map may degenerate at small scales, and when it doesn’t.
In this talk we will discuss similar issues for a parabolic analogue of almost splitting maps, in the context of the Ricci Flow, and present some new results regarding the existence and small scale behaviour of almost splitting maps in a Ricci flow with Type I curvature bounds. It turns out that, as in the case of harmonic almost splitting maps, an almost splitting map at a given scale remains an almost splitting map even at smaller scales, modulo linear transformations, provided that the flow remains sufficiently selfsimilar. Moreover, we will see that the possible degeneration of these linear transformations is controlled in a certain way by a pointed version of Perelman’s W-entropy.
Gerhard Huisken
Inverse mean curvature flow and geometric inequalities on 3-manifolds
The talk discusses inverse mean curvature flow in 3-manifolds satisfying lower bounds on their scalar curvature and Ricci- curvature. We discuss how the flow can be used to derive sharp geometric inequalities arising from such constraints.
Tom Ilmanen
Some speculations on the network flow
Bruce Kleiner
Mean curvature flow in R^3 and the Multiplicity One Conjecture
An evolving surface is a mean curvature flow if the normal component of its velocity field is given by the mean curvature. First introduced in the physics literature in the 1950s, the mean curvature flow equation has been studied intensely by mathematicians since the 1970s with the aim of understanding singularity formation and developing a rigorous mathematical treatment of flow through singularities. I will discuss progress in the last few years which has led to the solution of several longstanding conjectures, including the Multiplicity One Conjecture. This is joint work with Richard Bamler.
Luciano Mari
On Bernstein type theorems for minimal graphs under Ricci lower bounds
In this talk, we study solutions to the minimal hypersurface equation defined on a complete Riemannian manifold M. The qualitative properties of such solutions are influenced by the geometry of M, and one may expect results similar to those holding in Euclidean space provided that M has non-negative sectional or Ricci curvature. We focus on Ric non-negative, a case for which the analysis is subtler, especially because the lack of uniform ellipticity of the mean curvature operator makes comparison theory difficult to use. I will survey on recent splitting, Liouville and and half-space theorems obtained by the author in collaboration with G. Colombo, E.S. Gama, M. Magliaro and M. Rigoli. Techniques range from heat equation to potential theoretic arguments.
Luca Martinazzi
Variational Problems under topological constraints
The seminal work of Brezis-Coron for 2-dimensional harmonic maps introduces a sharp estimate that leads to the existence of minimisers of the Dirichlet energy in different homotopy classes. This had important consequences in the study of 3-dimensional harmonic maps, leading to the celebrated result of Rivière about the existence of everywhere discontinuous harmonic maps and the partial regularity result of Hardt-Lin-Poon for minimisers of the axially symmetric relaxed Dirichlet energy.
We will discuss what analogies, novelties and conjectures arise when following a similar path for the 1-dimensional half-harmonic map case and for the 4-dimensional Yang-Mills functional. The talk will be based on joint works with Ali Hyder (TIFR Bangalore) and Tristan Rivière (ETH Zurich).
Barbara Nelli
Minimal surfaces with infinite boundary value on 3-manifolds
In the sixties, H. Jenkins and J. Serrin proved a famous theorem about minimal graphs in the Euclidean 3-space with infinite boundary values. After reviewing the classical results, we show how to solve the Jenkins-Serrin problem in a 3-manifold with a Killing vector field. This is a joint work with A. Del Prete and J. M. Manzano.
Alessandra Pluda
Network flow: the charm of the (apparent) simplicity
The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature, and can be interpreted as the gradient flow of the length. In this talk, I will consider its natural generalization to networks that are finite unions of sufficiently smooth curves whose endpoints meet at junctions. I will list the many technical challenges one has to face to give a puctual description of this evolution.
Andreas Savas-Halilaj
Codimension two graphical MCF and isotopy problems
In this talk I will consider the graphical mean curvature flow of area decreasing maps between acomplete Riemannian manifold of dimension greater or equal than two and a complete Riemann surface of bounded geometry. I will show how this flow can be used to obtain results concerning the topological type of maps with small 2-dilation. The talk is based on works with K. Smoczyk and R. Assimos.
Michael Struwe
The prescribed curvature flow on the disc
For given functions f and j on the disc B and its boundary, we study the existence of conformal metrics g on B with prescribed Gauss curvature f and boundary geodesic curvature 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.
Peter M. Topping
Uniqueness questions in Ricci Flow
Uniqueness of Ricci flows given smooth initial data on closed manifolds has been known since the beginning of the theory. However, if one relaxes these constraints and allows non-compact underlying manifolds, and/or rough initial data, and/or smooth initial data that is attained in a weak sense, then the problem becomes rather subtle. I will survey some of what is known, and explain some recent progress joint with Hao Yin.
Guofang Wei
Fundamental Gap of convex domains in space forms & surface
The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schrödinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics. We will review many recent fantastic results for convex domains in space forms of constant curvature with Dirichlet boundary conditions, starting with the breakthrough of Andrews-Clutterbuck. Then we will present a very recent estimate for the convex domain in surfaces with positive curvature. The last result is joint with G. Khan, H. Nguyen, M. Tuerkoen.
Jonathan Zhu
Distance comparison principles for curve shortening flows
For closed curves evolving by their curvature, the theorem of Gage-Hamilton and Grayson establishes that an embedded curve contracts to a round point. An efficient proof was later found by Huisken, with improvements by Andrews-Bryan, which uses multi-point maximum principle techniques. We’ll discuss the use of these techniques in other settings, particularly for the long-time behaviour of curve shortening flow with free boundary.