## Conference Program

The conference starts on Monday, November 29, at about 2pm.
It ends on Friday, December 3, around noon.
Below you find the planned time schedule of the conference.
The titles and abstracts of the talks (in alphabetical order of the speakers) can be found below the time table.

Registration takes place in front of the Aula in the KG I, see also the directions.

Time Monday Tuesday Wednesday Thursday Friday
9 - 10 - J. Jost M. Ritoré M. Struwe T. Lamm
10 - 10:30 - --- Break ---
10:30 - 11:30 - A. Mondino C. Bär M. Simon G. Huisken
11:30 - 12:30 - A. Dall'Acqua A. Schikorra J. Scheuer
12:30 - 2:30 - ---- Lunch ---- ---- Lunch ---- ---- Lunch ----
1:30 - 2:30 Registration/Information -
2.30 - 3.30 T. Rivière R. Kusner Hike Y. Li
3.30 - 4 --- Break --- --- Break --- --- Break ---
4 - 5 N. Kapouleas E. Mäder-Baumdicker M. Müller
5 - 6 A. Malchiodi - -
7 - - Conference Dinner - -
 Spaces of metrics of positive scalar curvature on manifolds with boundary Christian Bär (University of Potsdam) Unlike for closed manifolds, the existence of positive scalar curvature (psc) metrics on connected manifolds with nonempty boundary is unobstructed. We study and compare the spaces of psc metrics on such manifolds with various conditions along the boundary: $$H \geq 0$$, $$H = 0$$, $$H > 0$$, $$II = 0$$, doubling, product structure. Here $$H$$ stands for the mean curvature of the boundary and $$II$$ for its second fundamental form. "Doubling" means that the doubled metric on the doubled manifold (along the boundary) is smooth and "product structure" means that near the boundary the metric has product form. We show that many, but not all of the obvious inclusions are weak homotopy equivalences. In particular, we will see that if the manifold carries a psc metric with $$H \geq 0$$, then it also carries one which is doubling but not necessarily one which has product structure. This is joint work with Bernhard Hanke.
 Elastic Flow with modulated Stiffness Anna Dall'Acqua (Ulm University) On planar closed curves with fixed prescribed length we consider a bending energy that depends on an additional density variable. The aim is to study the associated $$L^2$$-gradient flow. Using the fact that we are in the plane, we can use the tangent angle to reconstruct the curve. By this approach the flow equation is a parabolic system of second order with suitable Lagrange multipliers. These are needed to impose, on one side, the condition that the curves remain closed during the evolution, and, on the other, that the length as well as the total mass are kept fixed along the flow. We show that, if the initial datum is smooth enough, the solution exists globally in time and, possibly, discuss the convergence of the flow.
 Singularity structure and surgery in geometric evolution equations Gerhard Huisken (University of Tübingen) The lecture presents recent estimates on curvature and gradient of curvature near singularities of hypersurfaces evolving along non-linear functions of curvature. It is explained how these estimates are instrumental in classifying singularities and enabling surgery algorithms.
 The Dirichlet problem for minimal graphs in higher codimension Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Leipzig) The Dirichlet problem for the minimal surface equation is one of the classical problems of geometric analysis. It was extended to higher dimensions in a series of works, culminating in that of Jenkins and Serrin for meanconvex domains. For higher codimension, however, Lawson and Osserman identified essential difficulties. In this talk, which represents joint work with Qi Ding and Yuanlong Xin, I show that the Dirichlet problem is still solvable on meanconvex domains in higher codimension $$m$$ when the boundary data are sufficiently small in $$m-1$$ codimensions. In view of the Lawson-Osserman examples, such a perturbative result seems natural.
 Existence, index, and characterizations of minimal surfaces Nicos Kapouleas (Brown University) I will discuss some recent results and (depending on time) ongoing work and related open questions on constructions, index, nullity, characterizations, and classification of minimal surfaces. In particular I will discuss recent results in the article by me and P. McGrath ”Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers”, arXiv:2001.04240v2, results on the Lawson surfaces in the articles by me and D. Wiygul “The index and nullity of the Lawson surfaces $$\xi_{g,1}$$”, Camb. J. Math. 8 (2020), 363-405 and “The Lawson surfaces are determined by their symmetries and topology”, arXiv:2010.09371, existence results by me and D. Wiygul in “Free boundary minimal surfaces with connected boundary in the $$3$$-ball by tripling the equatorial disc”, to appear in J. Differential Geometry, and “Minimal surfaces in the three-sphere by desingularizing intersecting Clifford tori”, Mathematische Annalen (electronic at the moment), and existence results by me and M. Li in “Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc”, J. Reine Angew. Math. 776 (2021).
 Recent progress on the Canham and Willmore problems Rob Kusner (University of Massachusetts Amherst) We discuss the Willmore problem (minimizing the bending energy $$W=∫H^2$$ among smoothly embedded closed surfaces in $$R^3$$ of genus $$g$$) and the related Canham problem (prescribing the isoperimetric ratio $$v = \frac{36\pi \text{Vol}^2 }{ \text{Area}^3} ∈ (0,1)$$ as well): • Existence for the Canham problem (joint with Peter McGrath, using recent results of Andrea Mondino and Christian Scharrer) involves constructing a comparison surface with $$W$$ less than $$8\pi$$ and arbitrarily small isoperimetric ratio $$v$$, which we do by gluing $$g+1$$ small catenoidal bridges to the bigraph of a singular solution for the linearized Willmore equation $$\Delta(\Delta+2) φ = 0$$ on the $$(g+1)$$-punctured sphere $$S^2$$. • Some evidence for the conjecture that the Lawson surfaces with $$W$$ less than $$8\pi$$ solve the Willmore problem includes their $$W$$-stability (joint with Peng Wang, building on work of Nicos Kapouleas and David Wiygul), proven by showing their Area Jacobi operator $$\Delta+2+|A|^2$$ has a spectral gap between eigenvalues $$–2$$ and $$0$$. A few more things (of likely interest to Ernst Kuwert may be mentioned (if time permits)!
 Diffusive stability results for the harmonic map flow and related equations Tobias Lamm (Karlsruhe Institute of Technology) Abstract: TBA
 Huber's theorem for conformally compact manifolds Yuxiang Li (Tsinghua University) Let $$\Omega$$ be a domain of a closed manifold $$(M,g_0)$$ with $$\dim M>2$$. Let $$g=u^\frac{4}{n-2}g_0$$ be a complete metric defined on $$\Omega$$. We will show that $$M\setminus\Omega$$ is a finite set when $$\int_\Omega|Ric(g)|^\frac{n}{2}dV_g$$ < $$+\infty$$. Such a result is not true if we replace Ricci curvature with Scalar curvature. Next, we will discuss the properties of conformal metrics with $$\|R\|_{L^\frac{n}{2}}$$<$$+\infty$$ on a punctured ball of a Riemannian manifold , and give some geometric obstacles for Huber's theorem in this case.
 Geometry of complete minimal surfaces at infinity and the Willmore Morse index of their inversion Elena Mäder-Baumdicker (TU Darmstadt) A complete minimal surfaces with embedded planar ends can be compactified via inversions at spheres. The resulting surfaces are Willmore surfaces in Euclidean space. We explain how the Willmore Morse index of these surfaces of spherical type can be computed by studying unbounded Area-Jacobi fields on the corresponding minimal surface. Also several geometric properties of the minimal surfaces can be used to get results about the Willmore Morse Index of the inverted surface. An example is whether the asymptotic planes towards the ends meet at one point. This talk is based on joint work with Jonas Hirsch and Rob Kusner.
 Łojasiewicz inequalities near simple bubbles Andrea Malchiodi (Scuola Normale Superiore) We consider the $$H$$-functional for maps from closed surfaces into the three-dimensional Euclidean space. This is related to Wente’s inequality, and its conformal critical points parametrize surfaces with constant mean curvature. We analyze quantitatively the fact some universal lower bound for the $$H$$-energy of critical points is only attained on the sphere, deriving also a Łojasiewicz-type inequality in some energy range and showing a gap phenomenon in positive genus. This is joint work with Melanie Rupflin and Ben Sharp.
 Optimal Transport, weak Laplacian bounds and minimal boundaries in non-smooth spaces with Lower Ricci Curvature bounds Andrea Mondino (University of Oxford) The goal of the seminar is to report on recent joint work with Daniele Semola, motivated by a question of Gromov to establish a “synthetic regularity theory" for minimal surfaces in non-smooth ambient spaces. In the setting of non-smooth spaces with lower Ricci Curvature bounds: - We establish a new principle relating lower Ricci Curvature bounds to the preservation of Laplacian bounds under the evolution via the Hopf-Lax semigroup; - We develop an intrinsic viscosity theory of Laplacian bounds and prove equivalence with other weak notions of Laplacian bounds; - We prove sharp Laplacian bounds on the distance function from a set (locally) minimizing the perimeter: this corresponds to vanishing mean curvature in the smooth setting; - We study the regularity of boundaries of sets (locally) minimizing the perimeter, obtaining sharp bounds on the Hausdorff co-dimension of the singular set plus content estimates and topological regularity of the regular set. Optimal transport plays the role of underlying technical tool for addressing various points.
 Curvature minimization with perpendicular free boundary Marius Müller (University of Freiburg) This talk presents a joint work with Ernst Kuwert. We study the minimization of the square integrated second fundamental form $$\mathcal{ W}(f ) := \int_{\Sigma} |A|^2 d\mu_g$$ among all suitable immersions $$f : Σ \to \mathbb{R}^n$$ that are confined to some $$C^2$$-domain $$\Omega\subset\mathbb{R}^n$$ and intersect its boundary ∂Ω perpendicularly. We introduce a suitable varifold formulation for the problem, in particular for the perpendicular boundary condition. Once this is set, we can start to minimize. The most challenging part is to obtain uniform area bounds for minimizing sequences. We have found an optimal condition under which such bounds can be obtained: No plane may intersect $$\partial\Omega$$ perpendicularly. Thankfully, minimization is immediate if this condition is not fulfilled. Another challenge is to exclude vanishing of minimizing sequences. To overcome this, we study a reflection procedure for curvature varifolds with boundary. This will lead us to very fundamental questions about the notion of a curvature varifold.
 Pansu-Wulff shapes in the Heisenberg group Manuel Ritoré (University of Granada) A sub-Finsler metric in the Heisenberg group is a left-invariant norm in the horizontal distribution that induces a crystallographic perimeter. The minimizers of this perimeter under a volume constraint are the Pansu-Wulff shapes. In this talk I will discuss recent progress on existence, characterization and regularity of these minimizers.
 Area Variations under Lagrangian and Legendrian Constraints Tristan Rivière (ETH Zurich) Abstract: TBA
 Ricci flow of $$W^{2,2}$$ metrics in four dimensions (joint work with Tobias Lamm) Miles Simon (Otto von Guericke University of Magdeburg) In this talk we construct solutions to Ricci DeTurck flow in four dimensions on compact manifolds which are instantaneously smooth but whose initial  values are (possibly) non-smooth Riemannian metrics whose components, in smooth coordinates, belong to $$W^{2,2}$$ and are bounded from above and below. A Ricci flow related solution is constructed whose initial value is isometric in a weak sense to the initial value of the Ricci DeTurck solution.
 The mean curvature flow in null hypersurfaces and the detection of MOTS Julian Scheuer (Cardiff University) This talk is about the mean curvature flow in $$3$$-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. Under fairly mild conditions we obtain that for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a smooth local foliation of the past of an outermost MOTS.
 Hölder-Analysis of the Topology of the Heisenberg group Armin Schikorra (University of Pittsburgh) The Heisenberg groups are examples of sub-Riemannian manifolds homeomorphic to the Euclidean space -- but the homeomorphism is not bilipschitz. Their metric is derived from curves which are only allowed to move in so-called horizontal directions. When one considers approximation or extension problems for Sobolev maps into the Riemannian manifolds it is known that topological properties of the target manifold play a role. However, due to the homeomorphism, the topology of the Heisenberg group is the same as the Euclidean space. A notion of Hölder topology is needed. I will report on some progress (with Hajlasz) on some topological features of the Heisenberg group, in particular on an embedding question due to Gromov.
 Plateau flow Michael Struwe (ETH Zurich) We  propose a new heat flow for the Plateau problem, defined in terms of an equation rather than a variational inequality as in previous work of Chang-Liu, and preserving the smoothness of the data. The flow is either defined for all time and converges to a minimal surface spanning the given curve, or a singularity develops, and we analyze the possible singularities, in particular, for boundary curves which are graphs over the boundary of a strictly convex region in the plane.