We introduce a new family of mass-type invariants for time-symmetric initial data in spacetimes satisfying the Dominant Energy Condition, with positive cosmological constant Λ > 0. Starting from Green's functions for the p-Laplacian with 1 < p < 3, we construct monotone quantities whose constancy characterizes the de Sitter solution — the natural zero-mass reference geometry in this setting. Evaluating these quantities on the minimal boundary, we define the Polarized p-harmonic Mass and the p-harmonic Total Mass, and prove Positive Mass Theorems for both, with rigidity characterizing round hemispheres of sectional curvature Λ/3. We also explain why this potential-theoretic approach overcomes the fundamental obstructions that prevent a direct use of the Hawking mass and the Inverse Mean Curvature Flow in the compact setting. This talk is the first of two complementary presentations; Part II will address the limiting case p → 1⁺ and the associated 1-harmonic mass theory.

Some years ago, in joint work with Gianmichele Di Matteo, we introduced a local singularity analysis for the Ricci flow, studying curvature blow-up rates locally around singular points. After recalling some of the most important definitions and results, we build on this framework to study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound. We prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results obtained with Enders and Topping as well as with Mantegazza that relied on a global Type I assumption. In the second part of the talk, we adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to minus infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order. 

Surfaces of constant spacetime (or co-dimension 2) mean curvature (STCMC) have been shown to be abundant in the asymptotic end(s) of any asymptotically Minkowskian spacetime of non-vanishing mass (C.—Sakovich ’21). It has since been an open question how many STCMC-surfaces there are in Minkowski spacetime. We will explain why there are in fact many STCMC-surfaces in Minkowski spacetime. Our analysis is based on a characterization of local STCMC-foliations in relativistic initial data sets by Metzger—Peñuela which in turn goes back to a local CMC-foliation result in Riemannian manifolds by Ye. We will also briefly touch on the corresponding Riemannian result by Yau who argues for rigidity of co-dimension 2 constant mean curvature (CMC) surfaces in Euclidean 4-space. In particular, we will indicate how the stark difference between the Euclidean and Minkowskian results can be resolved.

Minimal surfaces date back to Euler and Lagrange and to the beginnings of the calculus of variations. The techniques developed in this area have played key roles in geometry and partial differential equations.

In three-dimensional Euclidean space, much is now known about the structure of minimal surfaces. In contrast, already in four-dimensional Euclidean space there are many minimal hypersurfaces, yet few structural results.  We will discuss some of the first structure results in dimension four and above.

Stability of a fixed point of a continuous dynamical system is a key property as it governs the flow of nearby small perturbations. In an infinite dimensional setting, this 

is particularly striking for solutions of geometric partial differential equations associated to a (geometric) flow such as the mean curvature flow or the Ricci flow. In this talk we will focus on expanding gradient Ricci solitons (also known as forward self-similar solutions) and we will explain an unstable manifold theorem whenever such a soliton is not stable in the spectral sense. In particular, we will show the existence of non self-similar solutions to Ricci flow coming out of cones. Joint work with Tristan Ozuch and Felix Schulze.

The geometry of manifolds with non-negative scalar curvature is well-understood. They arise naturally as umbilic initial data sets to the Einstein Equations from General Relativity which satisfy the dominant energy condition. In case the umbilicity assumption is dropped the picture is much less clear. Some results like the positive mass theorem extend to this general setting, some do not, and some are notoriously open like the Penrose conjecture. In this talk we show that the Geroch conjecture still holds for general initial data sets. 

The lecture describes joint work in progress with Carlo Sinestrari on the existence of a unique sweep-out from infinity to the horizon on asymptotically flat 3-manifolds of positive mass given by a solution of mean curvature flow. The sweep-out provides an analogue of the Newton potential and defines a center of mass compatible with the center of mass given by constant mean curvature surfaces. 

The mean curvature flow offers a powerful mechanism for evolving extrinsic geometries into canonical forms. However solutions are subject to finite-time singularity formation. A profound problem, with many potential applications, is to reveal the geometric information encoded in these singularities. I will discuss work with Huy Nguyen in which we construct a flow with surgery for submanifolds of higher codimension under a natural curvature pinching condition. The key new contribution is fine control on singularities, allowing for them to be removed, after which the flow can continue smoothly. The construction has topological consequences. 

We prove existence of Yamabe metrics on singular manifolds with conical points and conical links of Einstein type that include orbifold structures. We deal with metrics of generic type and derive a counterpart of Aubin’s classical result. The singular nature of the metric determines a different condition on the dimension, compared to the regular case. In dimension four, we can also find solutions of min-max type in presence of Z_2-orbifold points. Some perspectives for other singular manifolds will be discussed. This is joint work with Mattia Freguglia and Francesco Malizia.

This talk continues Part I by focusing on the limiting case p → 1⁺ of the p-harmonic mass theory. Via a formal limiting procedure, we introduce the Polarized 1-harmonic Mass, defined through the Inverse Mean Curvature Flow with outer obstacle, and prove a Positive Mass Theorem with rigidity characterizing the de Sitter hemisphere. We then extend the theory to Riemannian bands modelling initial data with both a black-hole horizon and a cosmological horizon, establishing a Riemannian Penrose Inequality for the 1-harmonic mass, with rigidity characterizing Schwarzschild–de Sitter solutions. Throughout, we highlight how the 1-harmonic mass recovers the correct mass parameter for Schwarzschild–de Sitter metrics, providing a robust notion of mass for spacetimes with Λ > 0.

Abstract: For interfaces evolving by mean curvature, translating solitons are of interest as potential singularity profiles for the flow. This talk will be a report on the status of the collapsed case.

We show that, as time t → ±∞, such solitons in ℝ3 converge to uniquely determined collections of planes, where a key issue is the anisotropy of (non-)removable singularities at the infinities. I will explain how to carry out the needed global analysis of the quasilinear PDEs with non-perturbative drift, via sharp non-standard elliptic decay estimates for the drift Laplacian that imply improvements on the Evans-Spruck and Ecker-Huisken gradient and curvature a priori estimates, as we restrict to solitons. One also exploits a link from potential theory of the Yukawa equation to heat flows with L^infty data. A structure theory follows, and with it geometric applications: the solitons decompose at infinity into standard regions asymptotic to planes or (titled) grim reaper cylinders.

Joint with E. S. Gama and F. Martín. 

This talk will be about defining metric notions of distances between spacetimes, based on a comparison of distances within spacetimes rather than their metric tensors. Such distances are needed, in particular, for studying the convergence of spacetimes to limits that are not necessarily smooth. The main difficulty in addressing this question is that spacetimes are not natural metric spaces. We will discuss how to convert a broad class of spacetimes—equipped with (canonical) time functions satisfying a two-sided Lipschitz condition—to definite metric spaces using the null distance of Sormani-Vega. It turns out that this class of spacetimes admits a natural bi-Lipschitz atlas consisting of the so-called Temple charts, where the causal structure is encoded by the null distance and the time function. With the Temple atlas at hand, various notions of intrinsic distances between spacetimes can be defined. We will discuss, in particular, the so-called timed-Hausdorff distance. This distance is definite—meaning it equals zero if and only if there is a time-oriented Lorentzian isometry between the spacetimes—satisfies the triangle inequality, and yields a related compactness theorem. The talk is based on joint work with Christina Sormani and Benjamin Meco.

Abstract: We consider smooth, closed manifolds, and (possibly non-smooth) Riemannian metrics $g,$ which satisfy  $h \leq g \leq ch$ for some constant $c$ and some smooth

Riemannian metric $h$. We assume that the first weak derivative of the metric $g$ in any smooth coordinate system exists, and that the square of this derivative is locally integrable. In the case that the distributional scalar curvature of Lee--LeFloch (2015) is not less than some constant C, we give various examples of further conditions which guarantee that there is a smooth family of approximating metrics, each with scalar curvature not less than C in the smooth sense. The smooth approximating metrics are constructed using the Ricci DeTurck flow. This is joint work with Florian Litzinger

Lagrangian mean curvature flow refers to the mean curvature flow of Lagrangian submanifolds. By maximum principle and monotonicity formula arguments, broad classes of initial data form type II singularities, and one can often characterize blow-up limits of the flow at singularities. However, examples of Lagrangian mean curvature flow singularities with precise dynamics have so far been lacking. We'll discuss a new construction of Lagrangian mean curvature flow solutions that provides examples with quantitatively precise finite-time singularity formation. This is joint work with Wei-Bo Su (National Central University).

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 from 2026. The result involves the so-called PIC1 curvature condition that I will explain from scratch during the talk. 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. The new theorem builds on distinct earlier work of Lee and the speaker, and of Deruelle-Schulze-Simon, that established the result with (different) additional hypotheses. Joint work with Alix Deruelle, Man Chun Lee, Felix Schulze and Miles Simon.

Curvature flows with global constraints, such as preservation of length, area or enclosed volume, arise as natural modifications of classical geometric evolutions (see, for instance, [2, 1]). These flows introduce nonlocal terms that significantly affect the analytical and geometric properties of the evolution. Many arguments developed for unconstrained flows, such as the curve shortening or mean curvature flow, rely on maximum principle arguments and comparison properties. In the constrained setting, however, these tools are often no longer directly applicable, leading to new challenges in the PDE analysis and requiring refined estimates and alternative monotonicity formulas. Additional difficulties appear when the ambient space has nontrivial curvature. In particular, curvature effects may influence the preservation of geometric properties along the flow, and in variable-curvature settings the evolution equations involve extra terms that must be carefully controlled. This highlights the crucial role of the ambient geometry in the analysis of constrained curvature flows, leading to a variety of qualitative behaviors. On the one hand, for Hadamard surfaces, which have negative (but not necessarily constant) Gaussian curvature, one can obtain strong control on the evolution. In this setting, we discuss recent results from our recent work [3] on area- and length-preserving curvature flows of embedded curves, including preservation of convexity, long-time existence, and curvature estimates. On the other hand, in positively curved ambient spaces such as the sphere, the situation is markedly different. It is known that the preservation of convexity and other geometric properties may fail in this setting, reflecting the influence of the ambient curvature [1]. Motivated by this contrast, we present a notion of h-convexity, inspired by previous work in hyperbolic geometry and further developed in recent contributions [4], which provides a suitable framework to recover monotonicity properties and derive robust geometric estimates in curved ambient spaces.

REFERENCES

[1] G. Huisken, The volume preserving mean curvature flow, J. Reine Angew. Math., 1987.

[2] M. E. Gage, On an area preserving evolution equation for plane curves, in D. M. DeTurck (ed.), Nonlinear Problems in Geometry, Contemporary Mathematics, vol. 51, AMS, 1986, pp. 51–62.

[3] S. Albert-Niclòs, E. Cabezas-Rivas, Constrained curvature flows on pinched Hadamard surfaces, arXiv:2604.13734.

[4] S. Pan, J. Scheuer, The quermassintegral inequalities for horo-convex domains in the sphere, arXiv:2512.12565, 2025.

It is well-known that Einstein metrics arise as critical points of the renormalized Einstein-Hilbert functional. The second variation of this functional is closely related to the so-called Einstein operator, which is a self-adjoint elliptic operator acting on symmetric 2-tensors. In the compact setting, we say that an Einstein manifold is stable if the Einstein operator is non-negative on symmetric 2-tensors that are traceless and divergence-free; this definition can also be extended to non-compact manifolds. The aim of this talk is to provide stability results for Einstein manifolds in a variety of settings: by imposing conditions on the Weyl tensor, we establish new stability criteria for compact, asymptotically hyperbolic (AH), and asymptotically locally Euclidean (ALE) Einstein manifolds. This talk is based on joint work with K. Kröncke.

Expanding Ricci solitons are widely regarded as canonical models for the formation of initial singularities in Ricci flows. A thorough understanding of their geometric and analytic properties is therefore of fundamental importance. In this talk, we focus on the partial differential equations arising from the Kähler–Ricci flow on gradient asymptotically conical expanding Kähler–Ricci solitons. We investigate questions of uniqueness and stability for solutions evolving on such backgrounds. As an application of the resulting uniform estimates, we construct infinitely many nontrivial Ricci breathers on expanding Kähler–Ricci solitons.

I will discuss sharp curvature inequalities and rigidity results for constant mean curvature type surfaces in both Riemannian and Lorentzian geometry. In the Riemannian setting, I will describe an extension of the Christodoulou–Yau inequality under a weaker stability condition, yielding rigidity in the equality case without symmetry assumptions. I will also briefly mention analogues in the hyperbolic and spherical settings. In the Lorentzian setting, I will introduce a notion of stability for spacetime constant mean curvature surfaces and present the corresponding sharp inequality and rigidity result under the dominant energy condition. This talk is based on the preprint arXiv:2603.16707.

Penrose's heuristic argument for the spacetime Penrose inequality is  fundamentally dynamical: the area of an apparent horizon should be compared  with the mass measured at infinity after the black-hole exterior has settled  down. In this talk, I will explain how this heuristic can be made precise without assuming convergence of the spacetime to Kerr. More precisely, I will describe a new proof of the inequality under a precise ``quasi-final-state hypothesis'', a late-time hypothesis which asks for  much less than convergence of the spacetime to Kerr.

The approach is new and formulated directly in spacetime. The main new geometric ingredient is the definition of a ``tangentially maximal'' hypersurface, carrying a foliation by spacelike spheres whose timelike mean curvature vanishes. We show that these hypersurfaces are governed by a quasilinear inward-parabolic PDE, and we develop the corresponding a priori theory and prove global existence. On these hypersurfaces, the spacetime Hawking mass reduces to the Riemannian Hawking mass, and the dominant energy condition gives nonnegative scalar curvature. This reduces the spacetime Penrose inequality to its Riemannian counterpart. Together with the horizon area law, this yields the desired inequality and gives a rigorous realization of Penrose's original dynamical heuristic.

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 < \infty$, and $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small $\varepsilon_0(n)>0$. It was shown by Simon (2002) that a Ricci--DeTurck flow solution $g(t)_{t \in (0,T)}$ related to $g_0$ exists for some $T=T(n,K_0)>0$.  If $g_0 \in L^2_{\mathrm{loc}}$ or $g_0 \in W^{1,2+2\sigma}_{\mathrm{loc}}$, $\sigma \in (0,\frac{1}{4})$, respectively, we show that $g(t) \to g_0$ in the $L^2_{\mathrm{loc}}$- or $W^{1,2+\sigma}_{\mathrm{loc}}$-sense, respectively. This is joint work with Miles Simon.

In this talk, we present a positive mass theorem for the volume-renormalized mass. The result is obtained via a new monotonicity formula, which holds along the level sets of the minimal Green’s function for the Laplace operator with a pole, in asymptotically hyperbolic 3-manifolds with scalar curvature greater than or equal to −6, satisfying an appropriate topological assumption. This talk is based on a joint work with Klaus Kröncke and Alan Pinoy.

Everything I discuss is joint work with P. M. Topping. In 2024, Topping & the author proposed the principle of delayed parabolic regularity for the Curve Shortening Flow (CSF), that if two disjoint planar CSFs bound an evolving annular region of fixed (small) area A, then after waiting for a time t > τ (just) after a magic time τ = τ(A) depending on only A, the regularity of one curve is controllable at time t in terms of the area A, the time t - τ since the magic time and the regularity of the other curve at time t. They also proved, under additional structural constraints, that their principle holds as stated.

In this talk I will explain, in the setting of the framework above for closed curves, that the so-called modulus, a conformal invariant of the annular region, strictly increases under the CSF. Topping & the author expect this monotonicity to play a role in establishing the proposal above for closed curves in full generality.

In this talk, we construct an exhaustive family of constant spacetime mean curvature surfaces for initial data sets close to the anti-de Sitter-Schwarzschild hyperboloid. In particular, we obtain such a foliation as the long time limit of the volume preserving spacetime mean curvature flow starting from the constant mean curvature foliation constructed by Neves-Tian (Geom. Funct. Anal., 2009). As an application, inspired by the definition proposed in the asymptotically Euclidean setting by Cederbaum-Sakovich (Calc. Var. PDE, 2021), we study the center of mass of an asymptotically hyperboloidal initial data set.

In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow $F: \Sigma^n \times [0, T) \to \mathbb{R}^{n+1,k}$ of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace $\mathbb{R}^{n+1,0}\subset \mathbb{R}^{n+1,k}$