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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.
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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).
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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.