We introduce new mass-type invariants for time-symmetric initial data sets in space-times with positive cosmological constant satisfying the Dominant Energy Condition, which implies a positive lower bound on the scalar curvature. Our construction is based on monotone quantities associated with p-harmonic functions on the manifold.
We prove a Positive Mass Theorem for these new invariants, whose equality case characterises the de Sitter space-time. Time permitting, we also explain how, in the limit as p tends to 1, these p-invariants give rise to a new geometric invariant, which we call 1-harmonic Mass. This invariant can be defined purely in terms of geometric quantities associated with the inverse mean curvature flow with outer obstacle. It satisfies both a Positive Mass Theorem and a Riemannian Penrose Inequality.
The talk is based on joint work with S. Borghini and L. Mazzieri.
Reporting on joint work with Dan Knopf I will show that there is an infinite dimensional family of ancient solutions that in backward time converge to one of the Böhm solitons. I will also discuss how these results can be extended to ancient solutions of the expanding Ricci flow that in backward time converge to expanding solitons. If time permits I will also discuss the analogous results for Mean Curvature Flow.
Nilpotency & curvature lower bounds
We discuss the structure of fundamental groups of closed manifolds with lower curvature bounds. In particular, we focus on the Fukaya–Yamaguchi Almost Abelian Conjecture and provide a counterexample under the nonnegative Ricci curvature assumption. This is joint work with A. Naber and D. Semola.
This work addresses the liquid drop model, introduced by Gamow in 1930 and Bohr–Wheeler in 1939, to describe the structure of atomic nuclei in nuclear physics. The problem involves finding a surface in three-dimensional space that is critical for a specific energy functional, balancing surface tension and nonlocal repulsion, subject to a volume constraint. Spherical solutions always exist and minimize the energy for sufficiently small volumes. However, for larger volumes, constructing non-minimizing critical points becomes more challenging. In this study, we present a new class of large-volume solutions, resembling “pearl collars” arranged along an axis in the shape of a large circle, with geometry close to Delaunay’s unduloids—surfaces of constant mean curvature. We also construct non-minimizing solutions with small mass that resemble two nearly identical spheres connected by a narrow neck.
This is collaboration with Mónica Musso, Andrés Zúñiga, and Rupert Frank
Optimal scaling-invariant regularity is the first key step in the study of free‑boundary problems. In the talk I will present a new technique that allows for obtaining it for several Bernoulli‑type free‑boundary problems. The proof does not requireany monotonicity formula and thus covers, at the same time, vectorial and higher‑order problems, also with varying side constraints. I will also show how the technique allows for obtaining optimal regularity for stationary points in twodimensions, answering a question of Kriventsov and Weiss.
I’ll present recent results and works in progress involving the inverse mean curvature flow in 3-manifolds endowed with metrics of nonnegative scalar curvature. I’ll discuss applications to quasi-local and global mass lower bounds, that will be shown to be stable
under uniform limits of the metric tensor. Finally, I will touch upon refined results in progress on the well posedness of the inverse mean curvature flow in such limit spaces, together with applications. The talk is based on joint works and projects with Antonelli, Benatti, Gatti, Mazzieri, Nardulli, Pluda and Pozzetta.
From recent progresses in the study of smooth and nonsmooth Lorentzian structures it emerges the need of a functional-analytic theory where, among other things, the relevant norms satisfy a reverse triangle inequality. Aim of the talk is to show that perhaps such a theory is possible.
We will discuss our recent classification of all noncollapsed singularities of the mean curvature flow in R4. This is joint work with Kyeongsu Choi, and also builds on our earlier collaborations with Beomjun Choi, Toti Daskalopoulos, Wenkui Du, Or Hershkovits and Natasa Sesum.
Complete embedded minimal surface with integrable Gauss curvature such as the plane and the catenoid are fundamental objects in geometry. In this talk, I will show that the asymptotic slope of such a surface is bounded from below in an optimal way by a systolic quantity called the neck-size. A consequence of this inequality is a new characterization of the catenoid purely in terms of its extrinsic properties. This result confirms a conjecture of G. Huisken and can be viewed as an analog in extrinsic geometry of the Riemannian Penrose inequality in mathematical relativity. The proof is based on an analysis of so-called minimal capillary surfaces, which are compact minimal surfaces that intersect a given complete embedded minimal surface with integrable Gauss curvature at a constant angle. This is joint work with M. Eichmair.
The Weyl energy on four-manifolds is a geometric functional related to the Chern-Gauss-Bonnet formula. Similarly to Willmore’s functional for surfaces embedded in the three-dimensional Euclidean space, it enjoys conformal invariance properties. We are interested in the interaction of the Weyl’s functionals of two manifolds under the operation of connected sum, showing conditions that decrease the total energy. Such estimates might be useful in understanding compactness properties of minimizing or critical families of metrics.
Multi-valued harmonic functions arise naturally in several different contexts, including in the description of higher codimension minimal submanifolds. Using insights that emerged originally in gauge theory I will describe new constructions of minimal submanifolds with intricate branching sets. This is joint work with Federico Francheschini and Paul Minter. More broadly, I will also discuss some of the emerging theory of multivalued harmonic functions and its connection to the so-called thin obstacle problem. Finally, if time permits, I will also describe new constructions of nonproduct minimizing submanifolds with cylindrical tangent cones, joint work with Greg Parker.
The talk introduces a framework for studying the geometry of lightlike (null) hypersurfaces in spacetimes that may be singular or lack smooth structure. Building on tools from optimal transport and modern Lorentzian geometry, we develop a flexible, synthetic language to describe such hypersurfaces without relying on the usual smoothness assumptions.
Within this framework, we propose a synthetic formulation of the Null Energy Condition, a central concept in general relativity governing how matter and energy influence the causal structure of spacetime. This condition plays a fundamental role in Penrose’s singularity theorem—predicting the formation of black holes and recognized by the 2020 Nobel Prize in Physics—as well as in Hawking’s area theorem, which underpins black hole thermodynamics.
Joint work with Fabio Cavalletti (Milan) and Davide Manini (Vienna).
The talk focuses on the recent proof of the energy identity for nonlinear harmonic maps. The energy identity conjecture is about the blow up which may occur for sequences of nonlinear harmonic maps, and the first part of the lecture will be a gentle introduction to the topic with the focus of making clear the statement. The last part of the talk will be a brief overview of the proof with a focus on the new ideas needed. Several new and rather interesting estimates appear in the process, and possibly one of the main points is the ability to approximate the blow up set (in neck regions) by a smooth T which solves an equation. Work discussed is joint with Daniele Valtorta.
A Riemannian manifold (𝑀, 𝑔) is Ricci–pinched if Ric ≥ 0 and there exists a constant 𝜀 > 0 such that Ric ≥ 𝜀R𝑔. Hamilton’s pinching conjecture states that a complete, connected, Ricci–pinched Riemannian 3−manifold must be compact or flat. This conjecture has been addressed by Chen and Zhu, Lott, and Deruelle-Schulze-Simon, with a complete proof ultimately provided by Lee and Topping, relying on Ricci flow. Huisken and Körber have achieved an alternative proof using extrinsic geometric flows with an additional volume growth condition. Their approach is based on the monotonicity of the Willmore functional along the inverse mean curvature flow in Ricci–pinched manifolds. Rather than using the inverse mean curvature flow, one could consider 𝑝−harmonic potentials with 𝑝 ∈ (1, 2] and replace the Willmore functional with a suitable proxy, still achieving the conjecture under the extra growth condition.
In the talk, I will highlight the similarities and differences between these two methods, describe a version of the result for manifold with boundary and provide a unified perspective on a broad family of monotonicity formulas in (non)linear potential theory and along the inverse mean curvature flow.
We consider solutions to Ricci flow with scaling invariant curvature control starting at the initial time from a metric cone. Asymptotically conical expanding gradient Ricci solitons give rise to such solutions, but it is not expected that any such flow starting from a metric cone is self-similar. Nevertheless, under suitable assumptions on the curvature we show that at spatial infinity any such solution is exponentially close to an expanding gradient Ricci soliton. As a geometric application we show that any non-compact PIC1 pinched manifold with positive asymptotic volume ratio is flat. This is one of the main ingredients in the recent result discussed in Peter Topping’s talk. Joint work with Alix Deruelle and Miles Simon.
We will discuss 4d ancient solutions asymptotic to a bubble sheet. We will discuss the behavior of scalar curvature at infinity on complete non-compact steady gradient Ricci solitons. In dimension four, we identify the two edges of the soliton and prove that the
scalar curvature decays at a linear rate away from these edges. We show that if the scalar curvature vanishes at infinity, then the asymptotic cone is a ray. In particular, our results apply to the four-dimensional flying wings constructed by Lai. We will also mention deriving precise asymptotics for 4d symmetric ancient closed solutions asymptotic to a bubble sheet.
Volume- or area-preserving curvature flows are well known modifications of the standard flows of immersed hypersurfaces driven by extrinsic curvatures (e.g. the mean curvature). The additional nonlocal term brings some additional difficulties, such as the failure of the avoidance property and other arguments based onthe maximum principle. On the other hand, for these constrained flows there is often a monotone quantity, e.g. the isoperimetric ratio, which is not available in the standard case. This has allowed to prove convergence results of convex hypersurfaces to a spherical profile, by relatively simple arguments, in cases where the corresponding result for the standard flow is unknown or even false.
In the talk we survey some of these convergence results for volume preserving flows, namely
(i) flow by nonlinear functions of the mean curvature (with M.C. Bertini),
(ii) flow by fractional mean curvature (with E. Cinti and E. Valdinoci),
(iii) flow of capillary surfaces with prescribed boundary angle condition (with L. Weng).
Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, was resolved recently using Ricci flow. In this talk I will explain a direct analogue of that result in all dimensions. One aspect of the work is a new lifting technique that allows us to handle manifolds that are collapsed at infinity; until now this could only be achieved in 3D via work of Lott. One new Ricci flow ingredient is a curvature estimate that could previously be derived only with the strong curvature positivity hypotheses required to invoke the Harnack inequality. The new theorem builds on earlier work of Lee and the speaker, and separately of Deruelle-Schulze-Simon. This latter work will be explained in the talk of Felix Schulze. Joint work with Alix Deruelle, Man Chun Lee, Felix Schulze and Miles Simon.
Expanders are a special class of solutions to mean curvature flow in which each later time slice is a rescaled (expanded) copy of an earlier one. They play an important role in understanding both the long-time behavior of the flow and the asymptotics of solutions emerging from cone-like singularities. In this talk, I will survey recent progress in the theory of expanders and discuss their applications to related geometric problems.
We describe how the maximum principle can be used to prove gradient estimates in the setting of a composite material with two almost touching inclusions. Part of this is joint work with Yanyan Li.