Alessandra Pluda

Luca BENATTI (Universität Wien)

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

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.

 

Esther CABEZAS-RIVAS (Universitat de Valencia)

Geometric Analysis meets Image Processing

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.

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Alessandro CARLOTTO (Università di Trento)

Non-persistence of strongly isolated singularities, and geometric applications

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 ≥ 2) and codimension. As a special case, we prove that for any n ≥ 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 4π^2 − ε (for any ε > 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.
 

Carla CEDERBAUM (Tübingen University)

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

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.

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Semyon DYATLOV (MIT)

Control of eigenfunctions on negatively curved manifolds

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.

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Mattia FOGAGNOLO (Università di Padova)

Strictly mean convex exhaustions and isoperimetric inequalities

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.
 

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Giada FRANZ (CNRS, Université Gustave Eiffel)

Unknottedness of free boundary minimal surfaces and self-shrinkers

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.

 

Lan-Hsuan HUANG (University of Connecticut)

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

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 sta-
tionary 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.
 

Gerhard HUISKEN (MFO Oberwolfach and Universität Tuebingen)

Geometric concepts for quasi-local mass

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.

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Debora IMPERA (Politecnico di Torino)

Isoperimetric Inequalities on Manifolds with Asymptotically Non-Negative Curvature

In this seminar, I will examine the validity of the isoperimetric inequality on manifolds with asymp-
totically 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 pres-
ence of a small amount of negative curvature.
The results presented are part of a joint project with Stefano Pigola, Michele Rimoldi, and Giona
Veronelli.
 

Stephen LYNCH (King’s College London)

Mean curvature flows of higher codimension

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).

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Rafe MAZZEO (Stanford University)

Families of minimal varieties with nonproduct cylindrical tangent cones

An old construction by Caffarelli-Hardt-Simon shows that any truncated minimal cone in Eu-
clidean 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.

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Lorenzo MAZZIERI (Università di Trento)

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

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 cos-
mological 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 con-
jecture 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 ad-
dress this issue and, if time allows, explore applications in characterizing the de-Sitter spacetime.

 

Ettore MINGUZZI (Universita’ di Pisa)

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

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 moti-
vate 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).

 

Marco POZZETTA (POLIMI)

A sharp spectral splitting theorem

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).

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Anna SAKOVICH (Uppsala Universitet)

The mass of weakly regular asymptotically hyperbolic manifolds

In mathematical general relativity, the notion of mass has been defined for certain classes of man-
ifolds that are asymptotic to a fixed model background. Typically, the mass is an invariant com-
puted 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 inte-
grals 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 when-
ever the later are defined, and we expect that the positivity can be proven. This is joint work with
Romain Gicquaud.

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Mariel SAEZ (Pontificia Universidad Catolica de Chile)

On the existence and classification of k-Yamabe gradient solitons

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.

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Julian SCHEUER (Universität Frankfurt)

Capillary Christoffel-Minkowski problems

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, Alek-
sandrov, 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.

 

Carlo  SINESTRARI (Università di Roma ”Tor Vergata”)

Volume preserving curvature flows in Euclidean and Riemannian spaces

Since the earlier studies of curvature flows of immersed hypersurfaces, the interest of the re-
searchers 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 conver-
gence 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 curva-
ture 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).
 

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Christina SORMANI (CUNYGC and Lehman College)

Introducing Various Notions of Distances between Space-Times

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.

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Peter TOPPING (University of Warwick)

Delayed parabolic regularity for curve shortening flow

Linear parabolic PDEs like the heat equation have well-known smoothing properties. In reasonable
situations we can control the Ck 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.

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Xuwen ZHU (Northeastern University)

Analysis of gravitational instantons

Gravitational instantons are non-compact Calabi-Yau metrics with L2 bounded curvature and are
categorized into six types. I will describe three projects on gravitational instantons including:
(a) Fredholm theory and deformation of the ALHtype;
(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.

 

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