Schedule for: 23w5049 - Non-Linear Critical Point Theory in Analysis and Geometry

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat. I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione. Joint work with Dorian Martino.

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.

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. A very natural question is the one of constructing FBMS (in a given ambient manifold) of a given topological type. In this talk, we will focus on one of the methods that have been employed so far to tackle this problem, that is Simon-Smith variant of Almgren-Pitts min-max theory. We will see how this method allows us to control the topology (i.e. genus and number of boundary components) of the resulting surface, and we will present several applications. Based on joint work with Mario Schulz.

In this joint work with X. Zhou, we prove that in the three dimensional sphere with a bumpy metric or a metric with positive Ricci curvature, there exist at least four distinct embedded minimal two-spheres. This confirms a conjecture of S. T. Yau in 1982 for bumpy metrics and metrics with positive Ricci curvature. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.

Anisotropic area, a generalization of the area functional, arises naturally in models of crystal surfaces. Due to the lack of a monotonicity formula, the regularity theory for its critical points, anisotropic minimal surfaces, is much more challenging than the area functional case. In this talk, I will discuss how one can overcome this difficulty and obtain a smooth anisotropic minimal surface for elliptic integrands in closed 3-dimensional Riemannian manifolds via min-max construction. This confirms a conjecture by Allard [Invent. Math.,1983] in dimension 3. The talk is based on joint work with Guido De Philippis and Antonio De Rosa.

Let $(M, g)$ be an arbitrary 3D complete noncompact Riemannian manifold with nonnegative Ricci curvature. It has long been conjectured that the Ricci flow has a complete nonnegative Ricci curvature solution $g(t)$ emerging from $g$. Lai (2021) recently proved the conjecture is true provided $g(t)$ is not required to be complete for $t > 0$. In this talk I will discuss some conditions which guarantee the completeness of Lai's solution for positive times. The talk is based on joint work with Adam Martens.

Zoll metrics are Riemannian metrics on compact manifolds whose non-trivial geodesics are periodic, simple and with the same fundamental period. They have been investigated systematically since the discovery by Otto Zoll of non-trivial rotationally symmetric examples in the Euclidean space, but their classification is still elusive. On the other hand, there are different characterizations of such metrics from geometric-variational perspectives, which lead to the curious possibility of investigating "the Zoll property" in the more general context of minimal submanifold theory, or even other variational theories. This talk will be about this circle of ideas, and will be based on some recent joint work with some of our collaborators (F. Marques, A. Neves, R. Montezuma and R. Santos).

It has been a long-standing conjecture by Thurston which asserts that every almost Fuchsian manifold is foliated by closed incompressible constant mean curvature (CMC) surfaces. In this talk I will discuss our recent work using the modified mean curvature flow to prove the existence of closed incompressible surfaces of constant mean curvature in a certain class of almost Fuchsian manifolds. As an application, we confirm Thurston’s CMC foliation conjecture for such a class of almost Fuchsian manifolds. This is joint work with Zheng Huang and Zhou Zhang.

In this talk we construct and classify convex ancient mean curvature flows in the unit ball with free boundary on the sphere. We will describe the ideas involved in this classification result in dimension one, the case of curve shortening flow, and at the end briefly discuss how they extend in any dimension. This work is joint with Mat Langford.

We present an integral approach to the classical Hessian estimates for the Monge-Ampere equation, originally obtained via a pointwise argument by Pogorelov. The monotonicity employed here results from a maximal surface interpretation of "gradient" graph of solutions in pseudo-Euclidean space.

We explore a gradient flow for volume on Hamiltonian isotopy classes of Lagrangian submanifolds. The flow is modeled locally by a fourth order quasilinear parabolic equation. We discuss short-time existence and highlight some similarities and differences with mean curvature flow.

I will introduce a fractional notion of k-dimensional measure, with $0 \leq k < n$, that depends on a parameter $\sigma$ that lies between 0 and 1. When $k = n−1$, this coincides with the fractional notions of area and perimeter, and when $k = 1$ this coincides with the fractional notion of length. We will see that, when multiplied by the factor $1 − \sigma$, this $\sigma$-measure converges to the k-dimensional Hausdorff measure up to a multiplicative constant. This is based on a joint work with Brian Seguin.

Over nearly all scientific organisations, across every country and across time one finds that the progression of women in research/academia is significantly hindered when compared to men. Such a universal truth represents an enormous loss of talent, including in our very own scientific communities. Recent years have seen some progress in understanding the principal factors behind this phenomenon and there has been some progress in new schemes which are designed to address the lack of women in senior scientific positions. These schemes have also met with some resistance which, in itself, has been revealing of the reasons why there is such a difference in the progression rates of men and women in science. This presentation discusses some of the evidence behind gender equality, and—most importantly—how that can be translated into day to day practice within an academic department.

In this talk we will focus on the qualitative properties of the solutions to the fractional Yamabe problem in the n-dimensional Euclidean space which present an isolated singularity. In particular, we will see that the Morse index of any such solution is infinity. The proof uses an Emden Fowler type transformation, so that we can pass to a nonlocal problem posed in the 1-dimensional Euclidean space. This is a joint work with Sergio Cruz-Blázquez and David Ruiz

The study of optimal upper bounds for Laplace eigenvalues on closed surfaces is a classical problem of spectral geometry. Its most fascinating feature is the connection between critical metrics and harmonic maps to the sphere, i.e. critical points of the energy functional. It appears that such a correspondence between critical points of spectral and geometric functionals is not uncommon as several other instances were found in the recent years. In the present talk we will discuss the example of Dirac eigenvalues on surfaces, whose variational theory turns out to be connected to a special class of harmonic maps to complex projective spaces. The talk is based on a joint work with I. Polterovich and A. Métras.

Refining the compactness theory jointly developed with Haslhofer, we develop a bubble-tree convergence result for sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds. This result implies a local energy identity for such a sequence and a local diffeomorphism finiteness theorem. Joint work with Louis Yudowitz.

We prove that every strictly convex 3-ball with nonnegative Ricci-curvature contains at least 3 embedded free-boundary minimal 2-disks for any generic metric, and at least 2 solutions even without genericity assumption. Our approach combines ideas from mean curvature flow, min-max theory and degree theory. We also establish the existence of smooth free-boundary mean-convex foliations. This is joint work with Dan Ketover.

Plateau's problem asks whether every boundary curve in 3-space is spanned by an area minimizing surface. Various interpretations of this problem have been solved using eg. geometric measure theory. Froehlich and Struwe proposed another approach, in which the desired surface is produced using smooth sections of a twisted line bundle over the complement of the boundary curve. The idea is to consider sections of this bundle which minimize an analogue of the Allen--Cahn functional (a classical model for phase transition phenomena) and show that these concentrate energy on a solution of Plateau's problem. After some background on the link between phase transition models and minimal surfaces, I will describe new work with Marco Guaraco in which we produce smooth solutions of Plateau's problem using this approach.

On the round sphere, the length of a closed geodesic is controlled by the area. We show that the same holds true for spherical surfaces with not too large Willmore energy. The energy gap we find is not only optimal but also exclusive to surfaces with genus zero. This is joint work with M. Müller (Leipzig/Freiburg) and C. Scharrer (Bonn).

In order to describe the behaviour of an elastic material undergoing fracture we can use a variational model and the so-called Mumford-Shah energy defined on a subspace of SBV functions. One difficulty is that the critical points of this energy are difficult to approximate by numerical methods. One can then think of approximating the Mumford-Shah energy by another energy defined on a space of more regular functions (H1-functions) : the Ambrosio-Tortorelli energy. It is known since the pioneer work of Ambrosio-Tortorelli that the minimizers of this energy converge towards minimisers of the Mumford-Shah energy. In this talk we will show that, under an assumption of convergence of the energies, critical points of the Ambrosio-Tortorelli energy also converge to critical points of the Mumford-Shah energy. This is a joint work with Jean-François Babadjian and Vincent Millot.

The conjecture will be stated immediately and then I will present our result: We prove that if $u : B_1 \subset \mathbb{R}^2 \to \mathbb{R}^n$ is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ is smooth. In particular this solves a conjecture of Lawson and Osserman from 1977 in the planar case. The remainder of the time I will use to present some ideas to the proof and how 2 dimensions enter. This is joint work with Connor Mooney and Riccardo Tione.

In this talk I will introduce Alexandrov immersed mean curvature flow and extend Andrew's non-collapsing estimate to include Alexandrov immersed surfaces. This implies a gradient estimate for the flow and allows mean curvature flow with surgery to be extended beyond flows of embedded surfaces to the Alexandrov immersed case. This is joint work with Elena Mäder-Baumdicker.

Regularity of minimizing p-harmonic maps – i.e., minimizers of the Dirichlet p-energy among maps between two given manifolds – is known to depend on the topology of the target manifold. In particular, the case of maps into spheres has been studied intensively, but still some of the most basic questions concerning maps from B3 into S3 remain open. Minimizing maps in this context were shown to be regular when $p=2$ or $p\geq 3$, and recently also when $2 < p < 2.13$, leaving a peculiar gap in between. I will discuss known approaches to the problem and how to prove regularity for $2 < p < 2.36$ and $2.97 < p <3$, thus shrinking the gap. This is joint work with Michał Miśkiewicz (University of Warsaw) and Andreas Gastel (University of Duisburg-Essen).

We consider the class of $m$-dimensional surfaces in a container with orthogonal boundary condition. The question whether the area is bounded in terms of the $L^p$ curvature leads to a varifold version. For $m = p = 2$ this also yields existence of a curvature minimizer in the class.