Gerhard Huisken --- Sweeping Out Asymptotically Flat 3-manifolds by Mean Curvature Flow
Asymptotically flat 3-manifolds appear as space-like time-slices of Lorentzian 4-manifolds in the description of isolated gravitating systems. We describe recent joint work with C. Sinestrari the construction of ancient solutions of mean curvature flow that sweep out the entire region between spatial infinity and the outermost horizons of black holes in the system, providing a natural geometric radial coordinate. The results rely on new a priori estimates for ancient solutions in the asymptotically flat setting.
Francisco Martin --- When Mean Curvature Flow Creates Topology: On a Conjecture of Ilmanen
In 1995, Tom Ilmanen conjectured that the genus of a surface evolving by mean curvature flow should decrease whenever a singularity occurs, with the exception of certain standard singularities. Although this principle holds for important classes of flows, we will show that it fails dramatically for general weak solutions.
More precisely, we construct self-similar mean curvature flows that pass through a conical singularity and emerge with arbitrarily large genus. In fact, a single asymptotic cone can arise from a self-shrinking surface of fixed genus and admit self-expanding continuations of every nonnegative genus.
The construction combines earlier work on high-genus self-shrinkers with new existence results for self-expanders. A central ingredient is an application of degree theory to spaces of minimal surfaces, together with symmetry arguments, deformations to classical periodic minimal-surface problems, and compactness techniques.
This is joint work with David Hoffman and Brian White.
Andre Neves --- Counting minimal Lagrangians via Mirzakhani’s function
I will explain how Mirzakhani’s count of simple closed geodesics can be used to count minimal Lagrangians on some Kahler-Einstein 4-manifolds. This is joint work with Marques and Lowe.
Lu Wang --- Relative expander entropy and its applications
Self-expanders play an important role in the analysis of mean curvature flow, particularly in understanding the resolution of conical singularities and the long-time behavior of the flow. In this talk, I will present recent results on relative expander entropy, focusing on its analytic and geometric properties. I will discuss how the relative expander entropy framework illustrates the structure of expanders and the role of monotonicity in understanding mean curvature flow.
Felix Schulze --- TBA
TBA
Bill Minicozzi --- Confinement and growth of minimal surfaces
Solutions of elliptic equations tend to spread out evenly and confining them forces strong restrictions. This appears in linear PDE globally with the Liouville theorem and locally with the Harnack inequality. We prove nonlinear versions of this for minimal varieties in any dimension or codimension. The first is a volume doubling theorem that implies polynomial growth in sufficiently narrow cones. This can be sharpened to Euclidean volume growth in sublinearly growing regions. Both results are optimal. This yields, among other consequences, Bernstein-type rigidity for confined stable hypersurfaces in high dimensions. This is joint work with Toby Colding; the examples are joint also with Fransisco Martin.
Robert Haslhofer --- Ancient cylindrical flows with fast or slow convergence
In a recent breakthrough, Bamler-Lai proved Ilmanen's mean-convex neighborhood conjecture in all dimensions by showing that any nontrivial ancient asymptotically cylindrical mean curvature flow is -- up to splitting Euclidean factors -- either a translating bowl, or an ancient oval, or a translating oval-bowl. In this talk, I will discuss joint work with Kyeongsu Choi, where we provide a short alternative argument for two of the three scenarios. Specifically, using more elementary/traditional methods, we show that if the convergence to the round cylinder is fast then the solution is a bowl times a Euclidean factor, and if the convergence is slow then the solution is an ancient oval. Moreover, our approach also yields a new proof of the mean-convex neighborhood conjecture for neck-singularities that substantially simplifies and streamlines the proof from my prior work joint with Choi-Hershkovits (Acta '22) and Choi-Hershkovits-White (Inventiones '22).
Michael Eichmair --- The Penrose inequality in extrinsic geometry
A classical result asserts that every complete embedded minimal surface of finite total curvature in $\mathbb{R}^3$ has finitely many ends, each asymptotic to a plane or to a catenoid. The flux of the topmost end is in some sense analogous to the mass of an asymptotically flat manifold. I will show that it is bounded below in terms of the least amount of extrinsic area needed to separate the topmost end from the rest of the surface with equality holding if and only if the minimal surface is a flat plane or the catenoid. This result is a special case of the Penrose inequality in extrinsic geometry as conjectured by Gerhard Huisken and proven recently in my joint work with Thomas Koerber.
Richard Bamler --- Ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture
We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons.
Our approach relies on two new techniques. The first, called the PDE-ODI principle, converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This framework bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. The second combines a new leading mode condition combined with an "induction over thresholds" argument" to obtain even finer asymptotic estimates.
This is joint work with Yi Lai.