Note: updated sporadically, and some talks (particularly job related talks) may be missing.

**Winter/Spring ’24 seminar:**

- Artemis Vogiatzi (QMUL)

Title: High Codimension Mean Curvature Flow in CP^n

Abstract: Mean curvature flow is a geometric evolution equation that describes how a submanifold embedded in a higher-dimensional space changes its shape over time. We establish a codimension estimate that enables us to prove at a singular time of the flow, there exists a rescaling that converges to a smooth codimension one limiting flow in Euclidean space, regardless of the original flow’s codimension. Under a cylindrical type pinching, we show that this limiting flow is weakly convex and either moves by translation or is a self-shrinker. These estimates allow us to analyse the behaviour of the flow near singularities and establish the existence of the limiting flow. Considering the ℂPn, we go beyond the finite timeframe of the mean curvature flow, by proving that the rescaling converges smoothly to a totally geodesic limit in infinite time. Our approach relies on the preservation of the quadratic pinching condition along the flow and a gradient estimate that controls the mean curvature in regions of high curvature.

- Jan Sbierski (Edinburgh)

Title: The C^0-inextendibility of a class of FLRW spacetimes.

Abstract: A Lorentzian manifold is said to be C^0-inextendible if there does not exist an isometric embedding into a bigger Lorentzian manifold of the same dimension with a continuous metric. I discuss recent work establishing the C^0-inextendibility of FLRW spacetimes without particle horizons in the case of positive spatial sectional curvature and, with some additional assumptions on the scale factor, in the case of negative spatial sectional curvature. Such inextendibility statements provide a classification for the strength of gravitational singularities in general relativity.

- Rodrigo Avalos (Tübingen)

Title: A Q-curvature positive energy theorem and rigidity of Q-singular manifolds.

Abstract: In this talk we will present recent results related to a notion of energy which is associated with fourth-order gravitational theories, where it plays an analogous role to that of the classical ADM energy in the context of general relativity. We shall show that this quantity obeys a positive energy theorem with natural rigidity in the critical case of zero energy. Furthermore, we will comment on how the resulting notion of energy underlies rigidity phenomena in geometry associated with Q-curvature, in particular, in the case of asymptotically Euclidean (AE) Q-singular spaces. These spaces contain Riemannian manifolds which satisfy a fourth-order analogue of the Einstein condition, which we refer to as J-Einstein manifolds. The analysis of J-Einstein manifolds is analytically more challenging, involving fourth order geometric partial differential equations, but several interesting properties of Einstein manifolds are still retained by this wider family. In this talk we shall show that the J-tensor retains optimal controls on the decay of the metric tensor at infinity, and also that J-flat Yamabe positive AE manifolds exhibit the same rigidity properties as Ricci-flat AE manifolds do.

- Stephen Lynch (Imperial)

Title: Singularities of fully nonlinear geometric flows.

Abstract: We will discuss the evolution of hypersurfaces by fully nonlinear parabolic geometric flows. Solutions to these flows typically form singularities in finite time. Understanding the kinds of singularities which can form is a necessary step towards many potential geometric applications. We will present a complete picture of the possible singularities which can form in low dimensions. The main results can be understood as Liouville-type rigidity theorems for certain concave/convex fully nonlinear parabolic PDE.

- Huy The Nguyen (QMUL)

Title: Mean Curvature Flow in the Sphere.

Abstract: In this talk, I will discuss some recent results analysing singularities of the mean curvature flow in the sphere both in the hypersurface case and the high codimension case.

- Louis Yudowitz (KTH)

Title: Dynamical Stability and Instability of Poincare-Einstein Manifolds.

Abstract: A lot of work over the past decade or so has been devoted to proving the stability of compact Ricci solitons (of which Einstein manifolds are a special case). These solitons evolve self-similarly under Ricci flow and arise as critical points of certain geometric functionals. On the other hand, the non-compact case still poses issues, in large part due to a lack of suitable functionals. In this talk, we will see how to overcome this for Poincaré-Einstein manifolds (i.e. negative Einstein manifolds which are asymptotically hyperbolic) and develop a complete picture of their dynamical stability. In particular, stability will depend on the metric being a local maximizer/minimizer for a suitable (relative) entropy functional for asymptotically hyperbolic manifolds, which was recently introduced by Dahl-McCormick-Kröncke. Along the way, we will also prove a Łojasiewicz-Simon inequality for this entropy, which will serve as our main technical tool. Local maximizers of the entropy will also be shown to be equivalent to a local positive mass theorem and a volume comparison result. This is all based on recent joint work with Klaus Kröncke.

**Fall ’23 seminar:**

- Dmitry Faifman (Tel Aviv)

Title: From uncertainty to rigidity in integral geometry.

Abstract: The Radon and cosine transforms are central to convex and integral geometry, in particular in geometric tomography and convex valuation theory. The range of those operators has been described by Gelfand-Graev-Rosu and Alesker-Bernstein in representation-theoretic terms, and – for the Radon transform – also through a PDE by John, Grinberg, Gonzalez and Kakehi.

We will discuss a rigidity phenomenon imposed on functions in these ranges, realized through restrictions on their support, or more precisely, a quasianalyticity property of those ranges in the sense of Kazdan. This will lead to the strengthening of classical theorems of Aleksandrov and Funk in geometric tomography, and of Klain and Schneider in convex valuation theory. The results are based on a novel support-type uncertainty principle on grassmannians.

- Darya Sukhorebska (Muenster)

Title: Simple closed geodesics on a surface with conical singularities.

Abstract: On a regular convex surface there are at least three simple closed geodesics. However this statement is not true for a non-smooth surface. The existence of a closed geodesic on such surfaces depends on the full angle around the singular point. A big challenge is that a geodesic that hits the singular point can not be extended. In my talk I will present a whole classification of simple closed geodesics on regular tetrahedra in spaces of constant curvature. We will also see the difference in properties of the geodesics that depends on the curvature of ambient space. I will also show you methods that we use for working with a non-smooth surface. The talk is based on a joint work with Alexander Borisenko.

- Franco Vargas Pallete (Yale)
**(part of a geotop geometry day)**

Title: Isoperimetric profile comparison for convex co-compact hyperbolic 3-manifolds

Abstract: Convex co-compact hyperbolic 3-manifolds are the generic infinite volume hyperbolic 3-manifolds with finitely generated fundamental group, while an isoperimetric profile is a function that describes the smallest perimeter to bound a region of a given volume. Using results of renormalized volume (which is a renormalization of the infinite volume) we will show a comparison and rigidity for isoperimetric profiles of convex co-compact hyperbolic 3-maniffolds and their model hyperbolic geometry. We will use this comparison and rigidity to prove a comparison and rigidity result for the Cheeger constant (optimal perimeter/volume ratio) on a class of convex co-compact manifolds. We will show that the Cheeger constant attains its maximum value on Fuchsian manifolds. This is based on joint works with Celso Viana.

- Shengwen Wang (Queen Mary U)
**(part of a geotop geometry day)**

Title: Phase transitions with Allen-Cahn mean curvature bounded in Lp.

Abstract: We consider the varifolds associated to a phase transition problem whose first variation of Allen-Cahn energy is Lp integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded Lp mean curvature. This is joint work with Huy Nguyen.

- Mario Schulz (Muenster)

Title: Topological control for min-max free boundary minimal surfaces

Abstract: Free boundary minimal surfaces naturally appear in various contexts, including partitioning problems for convex bodies, capillarity problems for fluids, and extremal metrics for Steklov eigenvalues on manifolds with boundary. Constructing embedded free boundary minimal surfaces is challenging, especially in ambient manifolds like the Euclidean unit ball, which only allow unstable solutions. Min-max theory offers a promising avenue for existence results, albeit with the added complexity of requiring control over the topology of the resulting surfaces. We establish general topological lower semicontinuity results for free boundary minimal surfaces obtained through min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We also present compelling applications, including the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.(Joint work with Giada Franz.)

**Winter/spring ’23:**

- Melanie Graf (Hamburg)

Title: Existence and uniqueness of ”nice” maximal boundaries of spacetime

Abstract: Questions of (low-regularity) spacetime (in-)extendibility have a long history within mathematical general relativity and are closely related to several important physical problems such as strong cosmic censorship and the nature of the incompleteness predicted from the singularity theorems. In case a spacetime is extendible at all it is therefore also essential to gain an understanding of how much uniqueness may be expected from purely geometric considerations (i.e., without assuming that the extension continues to obey a specific physical model or similar). Locally, this question was recently very comprehensively answered by Sbierski. In my talk I will present a possible approach to the global problem via considering certain extensions which are merely manifolds with boundary. After introducing the problem we will discuss that, under suitable assumptions on the regularity of the considered boundary extensions and excluding the existence of ”intertwined geodesics” in the original spacetime, which is known to lead to rather pathological situations, extendible spacetimes admit a unique maximal boundary extension. This is based on joint work with Marco van den Beld Serrano.

- Argam Ohanyan (Vienna)

Title: Non-smooth approaches to spacetime geometry

Abstract: Non-smooth spacetime geometry is a subject that has garnered a lot of interest in recent years. This is unsurprising, as even basic and physically well-motivated phenomena within smooth spacetime geometry lead to non-smooth scenarios. Another motivation is the success of non-smooth Riemannian geometry, where a study of metric length spaces and curvature bounds via triangle comparison has led to an incredibly fruitful theory which has delivered many new results even in the smooth context. In 2018, Lorentzian length spaces were put forth by Kunzinger and Sämann as the suitable synthetic setting for spacetime geometry.Since then, a lot of results from the classical theory of spacetimes have been rediscovered in this framework. In this talk, we first give the motivations for and discuss the basics of Lorentzian length spaces. We then continue with various recent developments related to curvature, Gromov-Hausdorff convergence and differential calculus in the synthetic

setting.

- Wei-bo Su (Warwick)

Title: Infinite-time singularities in Lagrangian mean curvature flow

Abstract: In this talk, I will present a gluing construction of infinite-time singularities in Lagrangian mean curvature flow. Starting with a pair of affine special Lagrangian tori $L = L_1\cup L_2$ intersecting transversely at a single point, we construct a long-time solution to Lagrangian mean curvature flow by perturbing the time-dependent Lagrangian surgeries $L_1\# L_2$ of $L$ at the intersection point parametrized by the “neck size” $\epsilon(t)$. Notably, the evolution of the “neck size” of the approximate solution is driven by the obstruction of the special Lagrangian deformation of $L_1\# L_2$. This talk is based on a joint work with Chung-Jun Tsai and Albert Wood.

- David Wiygul (ETH)

Title: Uniqueness and Morse index of minimal surfaces in the sphere and ball

Abstract: I will present some recent work with Alessandro Carlotto and Mario Schulz that in some respects builds on and in other respects complements earlier work with Nicos Kapouleas, which I will also review. Specifically, I will review the result with Kapouleas that each of Lawson’s embedded minimal surfaces in the 3-sphere is uniquely determined by its topological type and symmetry group, and I will review the calculation, also with Kapouleas, of the Morse index of an infinite subfamily of these same surfaces. Then, somewhat complementarily, I will describe the construction with Carlotto and Schulz of a family of free boundary minimal surfaces in the 3-ball having the same topological types and symmetries as those of a previously identified family, and I will further describe some index estimates for our new surfaces.

- Annachiara Piubello (Miami)

Title: Estimates on the Bartnik mass and their geometric implications

Abstract: In this talk, we will discuss some recent estimates on the Bartnik quasi-local mass for data with nonnegative Gauss curvature and positive mean curvature. This estimate is in terms of the area, the total mean curvature, and a quantity measuring the roundness of the metric. If the ratio between the maximum and the minimum of the Gauss curvature approaches 1, we show that our estimate converges to the sharp bound derived by Miao (2009) for round spheres with positive mean curvature. Furthermore, if the total mean curvature approaches 0, our estimate tends towards the sharp bound found by Mantoulidis and Schoen (2015) for the black hole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.

- Zhifei Zhu (YMSC, Tsinghua University)

Title: Systolic inequality on Riemannian manifold with bounded Ricci curvature

Abstract: In this talk, we show that the length of a shortest closed geodesic on a Riemannian manifold of dimension 4 with diameter D, volume v, and |Ric|<3 can be bounded by a function of v and D. In particular, this function can be explicitly computed if the manifold is Einstein. The proof of this result depends on a structural theorem proven by J. Cheeger and A. Naber. This is joint work with N. Wu.

- Sven Hirsch (Duke)

Title: On a generalization of Geroch’s conjecture

Abstract: The theorem of Bonnet-Myers implies that manifolds with topology $M^{n-1}\times S^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch’s conjecture shows that the torus $T^n$ does not admit a metric of positive scalar curvature. In this talk I will introduce a new notion of curvature which interpolates between Ricci and scalar curvature (so-called $m$-intermediate curvature) and use stable weighted slicings to show that for $n\le7$ the manifolds $M^{n-m}\times T^m$ do not admit a metric of positive $m$-intermediate curvature. This is joint work with Simon Brendle and Florian Johne.

- Aaron Chow (Columbia)

Title: Ricci flow and positivity of curvatures on manifolds with boundary

Abstract: In this talk, we would discuss how to construct a canonical solution to Ricci flow on manifolds with boundary given an arbitrary initial metric. This solution behaves well with respect to natural curvature conditions for Ricci flow on closed manifolds. Our construction has the advantage of preserving various important curvature conditions along the flow under natural assumptions on the boundary. As an application, this produces pleasant topological consequences for manifolds with boundary.

- Jingze Zhu (MIT)

Title: Spectral quantization for ancient asymptotically cylindrical flows

Abstract: Ancient asymptotically cylindrical flows are ancient solutions to the mean curvature flow whose tangent flow at $-\infty$ are shrinking cylinders. In this talk, we study quantized behavior of asymptotically cylindrical flows. We show that the cylindrical profile function u of these flows have the asymptotics $u(y,\omega, \tau) = \frac{y^{T}Qy – 2tr Q}{|\tau|} + o(|\tau|^{-1})$ as $\tau\rightarrow -\infty$, where $Q$ is a constant symmetric $k\times k$ matrix whose eigenvalues are quantized to be either 0 or $-\frac{\sqrt{2(n-k)}}{4}$. Assuming non-collapsing, we can further draw two applications. In the zero rank case, we obtain the full classification. In the full rank case, we obtain the $SO(n-k+1)$ symmetry of the solution. This is joint work with Wenkui Du.

- Rory Martin-Hagemeyer (UMass-Boston)

Title: On the Rate of Convergence of the Mean Curvature Flow

Abstract: We will prove that the rate of convergence of the rescaled mean curvature flow to a shrinker is at most exponential in the case of a compact initial hypersurface. We shall motivate and present the basic elements of the theory of mean curvature flow, starting with the basic elements of hypersurface geometry, shrinkers, rescaling of the mean curvature flow, and the convergence of the rescaled flow to a shrinker. Then we shall introduce a novel flow, the normal rescaled mean curvature flow, discuss its properties and compute some key evolution equations, and use it to establish our result.

- Peter McGrath (NCSU)

Title: Areas for genus zero free boundary minimal surfaces in the ball

Abstract: A *free boundary minimal surface (FBMS) *in the unit Euclidean 3-ball is a minimal surface whose boundary meets the unit sphere orthogonally. Such surfaces are critical points for the area functional among variations of the ball which preserve its boundary as a set. In surprising recent work, several authors have constructed sequences genus zero FBMS which converge to the unit sphere in the sense of varifolds. We show that the area of any genus zero FBMS is bounded above by $4\pi$, and show that the only way this inequality can be asymptotically achieved is by a sequence of surfaces converging to the sphere in the sense of varifolds. This is joint work with Jiahua Zou.