Participants and talks
Browse the talks and speakers who participated in our winter 2018 workshop at the Univesidad de Granada.
Compactness analysis for free boundary minimal hypersurfaces
The study of free boundary minimal hypersurfaces, namely of those critical points for the area functional that meet the boundary of their ambient manifold orthogonally, dates back at least to Courant and yet has recently seen remarkable developments both in terms of existence results (via min-max and or gluing/desingularization methods) and in terms of a much deeper understanding of the connection with extremal metrics for the first Steklov eigenvalue. In this lecture, I will survey a number of results, of global character, centered around the following general questions: What conditions ensure geometric compactness (i.e. smooth, graphical convergence with multiplicity one) of a sequence of free boundary minimal hypersurfaces? What is the fine scale description of the geometric picture when multiply-sheeted convergence happens instead? As a byproduct, our analysis allows to obtain various consequences in terms of finiteness and topological control, which we can combine with a suitable bumpy metric theorem in this category to derive novel genericity theorems. Based on joint work with Lucas Ambrozio and Benjamin Sharp.
Willmore surfaces and higher solutions of Hitchin’s self-duality equations
I consider surfaces in 3-space which are critical with respect to certain geometric variational problems, such as CMC and minimal surfaces and (constrained) Willmore surfaces. In this talk I want to report on recent results on the construction of new examples of higher genus CMC surfaces and on the identification of constrained Willmore minimizers in the class of conformal tori. Moreover, by viewing minimal surfaces in different space forms within the constrained Willmore integrable system, counterexamples to a question of Simpson are constructed. This suggests a deeper connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, and the rank 2 theory of Hitchin's self-duality equations. This talk is based on joint work with Cheikh Birahim Ndiaye, Sebastian Heller and Nicholas Schmitt.
Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-shrinkers of anisotropic mean curvature flow
We study a variational problem for surfaces in the euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered surface, which was introduced to model the surface tension of a small crystal. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium surfaces of this energy for volume-preserving variations are generalizations of the round sphere and constant mean curvature surfaces, respectively. However, they are not smooth in general. In this talk, we show that, if the energy density function is three times continuously differentiable and convex, then any closed stable equilibrium surface is a rescaling of the Wulff shape. Moreover, we show that, there exists a non-convex energy density function such that there exist closed embedded equilibrium surfaces with genus zero which are not (any homothety of) the Wulff shape. This gives also closed embedded self-similar shrinking solutions with genus zero of the anisotropic mean curvature flow other than the Wulff shape. These concepts and results are naturally generalized to higher dimensions.
Willmore minimizers with prescribed isoperimetric ratio
We discuss the existence of surfaces of type S2 minimizing the Willmore functional with prescribed isoperimetric ratio, and some asymptotics as the ratio goes to zero.
Characterization of f-extremal disks
A $f$-extremal domain in a manifold $M$ is a domain $\Omega$ which admits a positive solution $u$ to the equation $\Delta u+f(u)=0$ with $0$ Dirichlet boundary data and constant Neuman boundary data. Thanks to a result of Serrin, it is known that in $\mathbb R^n$ such a $f$-extremal domain has to be a round ball. In this talk, we will prove that a $f$-extremal domain in $\mathbb S^2$ which is a topological disk is a geodesic disk under some asumption on $f$. This is a joint work with J.M. Espinar.
Spherical cone metrics
I will report on recent progress on the problem of classifying metrics on surfaces with constant curvature and prescribed conic singularities, with particular emphasis on the case of positive curvature and large cone angles. Joint work with Xuwen Zhu.
William H. Meeks III
Progress in the theory of CMC surfaces in locally homogeneous 3-manifolds
I will go over some recent work that I have been involved in on surface geometry in complete locally homogeneous 3-manifolds X. In joint work with Mira, Perez and Ros, we have been able to finish a long term project related to the Hopf uniqueness/existence problem for CMC spheres in any such X. In joint work with Tinaglia on curvature and area estimates for CMC H>0 surfaces in such an X, we have been working on getting the best curvature and area estimates for constant mean curvature estimates in terms of their injectivity radii and their genus. It follows from this work that if W is a closed Riemannian homology 3-sphere W then the moduli space of closed stongly Alexandrov embedded surfaces of constant mean curvature H in an interval [a,b] with a>0 and of genus bounded above by a positive constant is compact. In another direction, in joint work with Coskunuzer and Tinaglia we now know that in complete hyperbolic 3-manifolds N, any complete embedded surface M of finite topology is proper in N if H is at least 1 (this is work with Tinaglia) and for any value of H less than 1 there exists complete embedded nonproper planes in hyperbolic 3-space (joint work with both researchers). In joint work with Adams and Ramos, we have been able characterize the topological types of finite topology surfaces that properly embed in some complete hyperbolic 3-manifold of finite volume (including the closed case) with constant mean curvature H; in fact, the surfaces that we construct are totally umbilic.
Polynomial conserved quantities for constrained Willmore surfaces
We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature surfaces in space-forms. We show that this hierarchy is preserved under both spectral deformation and Baecklund transformation, for special choices of parameters, defining, in particular, transformations of CMC surfaces into new ones, with preservation of both the space-form and the mean curvature, in the latter case. This is joint work with Susana Duarte Santos.
Minmax Hierarchies : A variational approach to the construction of Minimal Surfaces'
We introduce a general scheme that permits to generate successive min-max problems for producing critical points of higher and higher indices to Palais-Smale Functionals in Banach manifolds. We call the resulting tree of minmax problems a minmax hierarchy. Using the viscosity approach to the minmax theory of minimal surfaces that we introduced in a series of recent works, we shall explain how this scheme can be deformed for producing smooth minimal surfaces of strictly increasing area in arbitrary codimension. We shall implement this scheme to the case of the 3- dimensional sphere. In particular we are giving a min-max characterization of the Clifford Torus and conjecture what are the next minimal surfaces to come in the S3 hierarchy.
Minimal annuli in H2xR
We give some existence and non-existence results for properly Alexandrov-embedded minimal annuli in H2xR.
Discrete minimal and CMC surfaces, integrable systems and the Lawson correspondence
Minimal and constant mean surfaces have played a key role in classical (smooth) Riemannian geometry, but finding analogous discrete objects turns out to be difficult. Various definitions of these compete, e.g. critical points of the area functional, though that one remains unsatisfactory and breaks the maximum principle (as is obvious from the cotan Laplace operator). We will present here one based on circular quad-based nets and explain how it relates to the minimal/CMC PDE. We will also show that it does have an interpretation in terms of Lax pair, much like the smooth PDE has. As a consequence it offers a (partial) constructive approach, as well as a Lawson correspondence. This is a joint work with Alexander Bobenko (TU Berlin).
The geometry of overdetermined semilinear equations in the plane
Given a planar domain Ω and a function f(t) we consider bounded positive solutions of the overdetermined problem
Δu+f(u)=0 in Ω
u = 0 and ∂u/∂n = C on ∂Ω
If Ω is bounded, then by a classical result of Serrin, Ω is a disc and u is radial. If the domain is unbounded the shape of Ω and u are not so rigid. We present some theorems about that situation. These results show some relationship with the Geometry properly embedded minimal and constant mean curvature surfaces in R3. This is a joint work with David Ruiz and Pieralberto Sicbaldi.
Global solutions of Teichmüller harmonic map flow
We discuss the precise behaviour of solutions of Teichmüller harmonic map flow at finite time singularities and explain how it allow us to conclude that this geometric flow decomposes every given closed initial surface into (branched) minimal immersions. This is joint work with Peter Topping.
Complete minimal submanifolds of rank two in space forms
In this talk I will discuss several aspects of the geometric structure of complete minimal submanifolds of rank two in space forms. Under some natural curvature assumptions we provide a classification of these submanifolds. The results are joint work with M. Dajczer, Th. Kasioumis and Th. Vlachos.
A bubbling theory for minimal hypersurfaces
We’ll consider a theory of bubbling along a sequence of minimal hypersurfaces with bounded Morse index and area. In particular we’ll see that one can capture regions of coalescing index (along the sequence) to produce a family of complete minimal hypersurfaces in Euclidean space of finite total curvature (the ‘bubbles’). Furthermore the total curvature along the sequence is entirely accounted for by a limit minimal hypersurface and the bubbles. We’ll discuss applications of such a result. Joint work with R. Buzano and ongoing work with L. Ambrozio, R. Buzano and A. Carlotto.
Collapsing ancient solution of mean curvature flow
Understanding the geometry of ancient solutions for mean curvature flow is key to study singularities of mean curvature flow. In this talk, I will describe the construction of the unique compact convex rotationally symmetric ancient solution of mean curvature flow contained in a slab. This is joint work with Bourni and Langford.
Ricci flow and Ricci limit spaces
Ricci flow theory has been developing rapidly over the last couple of years, with the ability to handle Ricci flows with unbounded curvature finally becoming a reality. This is vastly expanding the range of potential applications. I will describe some recent work in this direction with Miles Simon that shows the right way to pose the 3D Ricci flow in this setting in order to obtain applications. Amongst these applications is a proof that 3D Ricci limit spaces are locally bi-Holder homeomorphic to smooth manifolds, which solves more than an old conjecture of Anderson-Cheeger-Colding-Tian in this dimension.
A Dichotomy Theorem for Minimal Surfaces
I will discuss a surprising dichotomy for classical minimal surfaces that gives new insights into the Colding-Minicozzi theory.