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
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.
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...
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...
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....
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....
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...
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...
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...
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...
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...
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,...
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...
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...
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....
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.
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...
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...
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...
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....
