Geometric concepts for quasi-local mass

• Huisken, Gerhard (MFO Oberwolfach and Universität Tuebingen)

Room: Aula Dini Several different notions of quasi-local mass for a region in a 3-manifold have been put forward. The lecture discusses some of their properties and deficiencies in the context of Geometric Analysis and Mathematical Relativity

A sharp spectral splitting theorem

• Pozzetta, Marco

Room: Aula Dini We present a splitting theorem for Riemannian manifolds that satisfy a spectral lower bound on the Ricci curvature. More precisely, given a Riemannian manifold with two ends, consider a Schrodinger operator whose potential is pointwise equal to the least eigenvalue of the Ricci tensor; we prove that if the spectrum of such operator is nonnegative, then the manifold has nonnegative Ricci in the pointwise classical sense and it splits isometrically. We will also discuss the sharpness of the assumptions. The result provides a sharp spectral generalization of the celebrated Cheeger-Gromoll splitting theorem in the case of multiple ends. The talk is based on a joint work in collaboration with Gioacchino Antonelli (New York University) and Kai Xu (Duke University).

Unknottedness of free boundary minimal surfaces and self-shrinkers

• Franz, Giada (CNRS, Université Gustave Eiffel)

Room: Aula Dini Lawson in 1970 proved that minimal surfaces in the three-dimensional sphere are unknotted. In this talk, we discuss unknottedness of free boundary minimal surfaces in the three-dimensional unit ball and of self-shrinkers in the three-dimensional Euclidean space. This is based on joint work with Sabine Chu.

Control of eigenfunctions on negatively curved manifolds

• Dyatlov, Semyon (MIT)

Room: Aula Dini Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. They have a long history of study going back to the Quantum Ergodicity theorem and the Quantum Unique Ergodicity conjecture. I will speak about the work with Jin and Nonnenmacher, proving that on a negatively curved surface, every semiclassical measure has full support. I will also discuss applications of this work to control for the Schrödinger equation and decay for the damped wave equation. Our theorem was restricted to dimension 2 because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and myself), was only known for subsets of the real line. I will talk about more recent joint work with Athreya and Miller in the setting of complex hyperbolic quotients and the work in progress by Kim and Miller in the setting of real hyperbolic quotients of any dimension. In these works there are potential obstructions to the full support property which can be classified by Ratner theory and geometrically described in terms of certain totally geodesic submanifolds. Time permitting, I will also mention a recent counterexample to Quantum Unique Ergodicity for higher-dimensional quantum cat maps, due to Kim and building on the previous counterexample of Faure-Nonnenmacher-De Bièvre.

On the constancy of surface gravity (temperature) for compact null hypersurfaces

• Minguzzi, Ettore (Universita' di Pisa)

Room: Aula Dini In Lorentzian geometry, a generic compact null hypersurface may not admit a tangential lightlike geodesic vector field. A classical conjecture (Isenberg-Moncrief) states that a lightlike pregeodesic vector field of constant surface gravity can be found in the totally geodesic case, provided suitable energy or convergence conditions are imposed. Surface gravity is significant due to its physical interpretation as temperature, particularly in the context of black hole physics. In this talk I motivate interest in this problem. Moreover, I explain how riemannian flow theory has helped classify the topology and flow structures of horizons independently of non-degeneracy assumption (e.g. assumptions on the completeness of generators)

Analysis of gravitational instantons

• Zhu, Xuwen (Northeastern University)

Room: Aula Dini Gravitational instantons are non-compact Calabi-Yau metrics with $L^2$ bounded curvature and are categorized into six types. I will describe three projects on gravitational instantons including: (a) Fredholm theory and deformation of the ALH* type; (b) non-collapsing degeneration limits of ALH* and ALH types; (c) existence of stable non-holomorphic minimal spheres in some ALF types. These three projects utilize geometric microlocal analysis in different singular settings. Based on works joint with Rafe Mazzeo, Yu-Shen Lin and Sidharth Soundararajan.

Introducing Various Notions of Distances between Space-Times

• Sormani, Christina (CUNYGC and Lehman College)

Room: Aula Dini We introduce the notion of causally-null-compactifiable space-times which can be canonically converted into a compact timed-metric-spaces using the cosmological time of Andersson-Howard-Galloway and the null distance of Sormani-Vega. We produce a large class of such space-times including future developments of compact initial data sets and regions which exhaust asymptotically flat space-times. We then present various notions of intrinsic distances between these space-times (introducing the timed-Hausdorff distance) and prove some of these notions of distance are definite in the sense that they equal zero iff there is a time-oriented Lorentzian isometry between the space-times. These definite distances enable us to define various notions of convergence of space-times to limit space-times which are not necessarily smooth. Many open questions and conjectures are included throughout. This is joint work with Anna Sakovich. (presenting online)

Non-persistence of strongly isolated singularities, and geometric applications

• Carlotto, Alessandro (Università di Trento)

Room: Aula Dini In this lecture, based on recent joint work with Yangyang Li (University of Chicago) and Zhihan Wang (Cornell University), I will present a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension. As a special case, we prove that for any $n\geq 2$ and any compact $(n+1)$-dimensional manifold $M$ the following holds: for a generic choice of the background metric $g$ all stationary integral $n$-varifolds in $(M,g)$ will either be entirely smooth or have at least one singular point that is not strongly isolated. In other words, for a generic metric only ``more complicated'' singularities may possibly persist. This implies, for instance, a generic finiteness result for the class of all closed minimal hypersurfaces of area at most $4\pi^2-\varepsilon$ (for any $\varepsilon>0$) in nearly round four-spheres: we can thus give precise answers, in the negative, to the well-known questions of persistence of the Clifford football and of Hsiang's hyperspheres in nearly round metrics. The aforementioned main regularity result is achieved as a consequence of the fine analysis of the Fredholm index of the Jacobi operator for such varifolds: we prove on the one hand an exact formula relating that number to the Morse indices of the conical links at the singular points, while on the other hand we show that the same number is non-negative for all such varifolds if the ambient metric is generic.

Delayed parabolic regularity for curve shortening flow

• Topping, Peter (University of Warwick)

Room: Aula Dini Linear parabolic PDEs like the heat equation have well-known smoothing properties. In reasonable situations we can control the $C^k$ norm of solutions at time $t$ in terms of $t$ and a weak norm of the initial data. This idea often carries over to nonlinear parabolic PDEs such as geometric flows. In this talk I will discuss a totally different phenomenon that can occur in some natural situations, in which there is an explicit magic positive time before which we have no regularity estimates at all, but after which parabolic regularity is switched on and we obtain full regularity. I plan to focus on the case of curve shortening flow, which will mean that almost no prerequisites will be assumed. Joint work with Arjun Sobnack.

The mass of weakly regular asymptotically hyperbolic manifolds

• Sakovich, Anna (Uppsala Universitet)

Room: Aula Dini In mathematical general relativity, the notion of mass has been defined for certain classes of manifolds that are asymptotic to a fixed model background. Typically, the mass is an invariant computed in a chart at infinity, which is related to the scalar curvature and has certain positivity properties. When the model is hyperbolic space, under certain assumptions on the geometry at infinity one can compute the mass using the so-called mass aspect function, a function on the unit sphere extracted from the term describing the leading order deviation of the metric from the hyperbolic background. This definition of mass, due to Xiaodong Wang, is a particular case of the definition by Chruściel and Herzlich which proceeds by taking the limit of certain surface integrals and applies to asymptotically hyperbolic manifolds with less stringent asymptotics. It turns out that these two approaches can be unified in such a way that the resulting definition of mass applies to asymptotically hyperbolic manifolds of very low regularity. In particular, in this setting one can use cut-off functions to define suitable replacements to the potentially ill-defined surface integrals of Chruściel and Herzlich. Moreover, the mass aspect function can be interpreted as a distribution on the unit sphere for metrics having slower fall-off towards hyperbolic metric than those originally considered by Xiaodong Wang. The related notion of mass is well-behaved under changes of coordinates and coincides with the notions of Wang, and Chruściel and Herzlich whenever the later are defined, and we expect that the positivity can be proven. This is joint work with Romain Gicquaud.

On the positive mass problem for initial data with a positive cosmological constant

• Mazzieri, Lorenzo (Università di Trento)

Room: Aula Dini The concept of mass for time-symmetric initial data has been extensively explored and is now a cornerstone in the study of contemporary Mathematical General Relativity, especially in relation to spacetimes with zero or negative cosmological constants. However, the case of a positive cosmological constant presents a distinct challenge, as our understanding is still unsatisfactory at the present stage. The renowned counterexample by Brendle, Marques, and Neves to the Min-Oo conjecture highlights that even the rigidity statement in a potential positive mass theorem has not been correctly identified yet in this context. In this presentation, I will propose approaches to address this issue and, if time allows, explore applications in characterizing the de-Sitter spacetime.

Combining potential theory with general relativity: a divergence theorem-based approach to proving geometric inequalities

• Cederbaum, Carla (Tübingen University)

Room: Aula Dini In 1977, Robinson gave a new proof of Israel’s celebrated static vacuum black hole uniqueness theorem by applying the divergence theorem to a very clever divergence identity based on the Cotton tensor from conformal geometry. His approach has found applications in several branches of general relativity and has inspired the analysis of Ricci solitons and quasi-Einstein manifolds. In this talk, we will demonstrate how one can combine his approach with linear and non-linear potential theory to give new proofs of classical as well as recent geometric inequalities such as the Willmore inequality in Euclidean space and its generalization to Riemannian manifolds with non-negative Ricci and Euclidean volume growth by Agostiniani—Fogagnolo—Mazzieri, the Minkowski inequality, and some geometric inequalities in general relativity. We will also show the relation to the corresponding classical potential theoretic approaches to these inequalities studied by Agostiniani, Fogagnolo, and Mazzieri. The results we will present are based on joint works with Florian Babisch, Albachiara Cogo, Benedito Leandro, Ariadna León Quirós, Anabel Miehe, and João Paulo dos Santos.

Nonlinear potential theory through the looking-glass and the Penrose inequality we found there.

• Benatti, Luca (Universität Wien)

Room: Aula Dini The Riemannian Penrose inequality states that the total mass of a time-symmetric spacetime is at least as large as the mass of the black holes it contains. Among the various proofs of this inequality, two are based on monotonicity formulas coming from distinct theoritical frameworks: one by Huisken and Ilmanen employing the inverse mean curvature flow, and another by Agostiniani, Mantegazza, Mazzieri, and Oronzio grounded in nonlinear potential theory. However, both rely on stronger assumptions than those required by the formulation of the inequality. In this talk, I will present a unified view that connects these two approaches. The monotonicity of the Hawking mass can be seen as the mirror image of a family of monotone quantities in potential theory: the two sides reflect and complete each other. This perspective allows us to extend the validity of the inequality to more general settings. This talk is based on joint work with M. Fogagnolo, L. Mazzieri, A. Pluda, and M. Pozzetta.

Geometric Analysis meets Image Processing

• Cabezas-Rivas, Esther (Universitat de Valencia)

Room: Aula Dini We study existence, uniqueness, and regularity of minimizers for a manifold-constrained version of the Rudin-Osher-Fatemi model for image denoising, which appears in multiple references of applied literature, but lacks analytical foundations. This leads to study a system of elliptic PDEs with Neumann boundary conditions. Our outcomes can be regarded as the extension to the harder situation of p=1 of the regularity theory for p-harmonic maps, started by classical works of Eells-Sampson and Schoen-Uhlenbeck. In fact, we generalize the optimal regularity results for the classical Euclidean scalar model, without further requirements on the convexity of the boundary, in three different directions: vector-valued functions, manifold-constrained and curved domain. To achieve the results, it is crucial on a strong interplay between geometric and analytical techniques within the proofs. Additionally, we provide variants of the regularity statement of independent interest: for 1-dimensional domains (related to signal denoising), local Lipschitz regularity (meaningful for image processing) and Lipschitz regularity for a perturbed model coming from fluid mechanics. This is joint work with Salvador Moll and Vicent Pallardó-Julià.

Capillary Christoffel-Minkowski problems

• Scheuer, Julian

Room: Aula Dini The classical Minkowski problem asks for the existence and uniqueness of a convex body with prescribed Gauss curvature, while the family of Christoffel-Minkowski problems generalize this question to find convex bodies with prescribed elementary symmetric polynomial of the principal radii. The full resolution of the Minkowski problem was given by works of Minkowski, Aleksandrov, Pogorelov, Nirenberg, Cheng-Yau, while sufficient conditions for the resolution of the Christoffel-Minkowski problem were given by Guan-Ma and Sheng-Trudinger-Wang. In this talk we discuss recent work with Yingxiang Hu and Mohammad Ivaki, which gives an analogous set of sufficient conditions to solve the Christoffel-Minkowski problem in the class of capillary surfaces in a half spaces with angle less than 90 degrees.

Isoperimetric Inequalities on Manifolds with Asymptotically Non-Negative Curvature

• Impera, Debora (Politecnico di Torino)

Room: Aula Dini In this seminar, I will examine the validity of the isoperimetric inequality on manifolds with asymptotically non-negative sectional curvature. After an overview of the current state of the art on the problem, I will discuss the ABP method, originally introduced by Cabré to prove the isoperimetric inequality in Rn, and later refined by Brendle for the case of non-compact Riemannian manifolds characterized by non-negative Ricci curvature and positive asymptotic volume ratio. Finally, we will explore how this approach can be adapted to prove isoperimetric inequalities even in the presence of a small amount of negative curvature. The results presented are part of a joint project with Stefano Pigola, Michele Rimoldi, and Giona Veronelli.

Volume preserving curvature flows in Euclidean and Riemannian spaces

• Sinestrari, Carlo (Università di Roma "Tor Vergata")

Room: Aula Dini Since the earlier studies of curvature flows of immersed hypersurfaces, the interest of the researchers has also been attracted by the volume preserving case, where the speed includes an additional nonlocal term which keeps the enclosed volume constant. For such flows it is usually possible to find a monotone quantity, e.g. the isoperimetric ratio, which is not available in the standard case. On the other hand, the nonlocal term induces the failure of some arguments based on the maximum principle, such as the avoidance property. Volume preserving flows have been studied in the past to show convergence of suitable classes of initial data to a spherical profile in the Euclidean setting, resp. to a CMC profile in the Riemannian case. Here we report on two recent developments along these lines. We first describe the convergence to a spherical cap of capillary surfaces with prescribed boundary angle condition under a general power mean curvature flow (joint work with L. Weng). We then consider the mean curvature evolution of large Euclidean coordinate spheres in asymptotically flat 3-manifolds of General Relativity, which allows to construct a CMC-foliation by extending a method of Huisken-Yau (joint wok with J. Tenan).

On the existence and classification of k-Yamabe gradient solitons

• Saez, Mariel (Pontificia Universidad Catolica de Chile)

Room: Aula Dini The k-Yamabe problem is a fully non-linear extension of the classical Yamabe problem that seeks for metrics of constant k-curvature. In this talk I will discuss this equation from the point of view of geometric flows and provide existence and classification results for soliton solutions of the k-Yamabe flow in the positive cone. This is joint work with Maria Fernanda Espinal.

Families of minimal varieties with nonproduct cylindrical tangent cones

• Mazzeo, Rafe (Stanford University)

Room: Aula Dini An old construction by Caffarelli-Hardt-Simon shows that any truncated minimal cone in Euclidean space lies in a large family of minimal submanifolds with isolated conic singularities, the elements of which are not exactly conic. It has been a long-standing open question to carry out some analogue of this construction for minimal varieties with nonisolated `cylindrical’ singular sets. I will discuss a new construction, which is a joint project with Greg Parker, where we construct deformation families of this nature. As has been suspected for some time, this turns out to be quite delicate from an analytic perspective because of an inherent rigidity, which can be recast as a loss of regularity in an iterative solution scheme. This result provides some support for the conjecture that the singular set of a minimal (minimizing?) subvariety must itself be a smooth submanifold once its regularity is better than some threshold.

​Mean curvature flows of higher codimension

• Lynch, Stephen (King's College London)

Room: Aula Dini Many fascinating phenomena occur when a submanifold of higher codimension is evolved by its mean curvature vector. In this more general setting much of the structure of hypersurface flows is absent e.g. embeddedness and mean-convexity fail to be preserved. Consequently, even in the simplest settings (closed curves in 3-space, surfaces in 4-space) basic questions remain unanswered. I will describe some of these open questions, and recent developments concerning flows satisfying natural curvature pinching conditions (from joint works with Nguyen and Bourni, Langford).

Causal Character of Killing Vectors and the Geometry of ADM Mass Minimizers

• Huang, Lan-Hsuan (University of Connecticut)

Room: Aula Dini We address two problems concerning ADM mass minimization for asymptotically flat initial data sets. First, we prove that data sets with zero ADM mass satisfying the dominant energy condition must embed in a pp-wave spacetime, without assuming spin. Second, we confirm Bartnik’s stationary vacuum conjecture for positive Bartnik mass. A key ingredient is a monotonicity formula for the Lorentzian length of a Killing vector field, together with a strong maximum principle. This is based on joint work with Sven Hirsch.

Strictly mean convex exhaustions and isoperimetric inequalities

• Fogagnolo, Mattia (Università di Padova)

Room: Aula Dini I will illustrate a general principle, first devised by Kleiner in a special case, allowing to pass from mean curvature inequalities to isoperimetric inequalities in general noncompact Riemannian manifolds that can be exhausted by strictly mean convex domains. I will then describe a trichotomy theorem envisioned by Gromov allowing one to decide whether a manifold enjoys such property or not. The talk is based mainly on works in collaboration with Borghini, Mazzieri and Santilli.