Globally hyperbolic spacetimes: slicings, boundaries and counterexamples

The Cauchy slicings for globally hyperbolic spacetimes and their relation with the causal boundary are surveyed and revisited. Our study covers: (1) adaptative possibilities and techniques for their Cauchy slicings, (2) global hyperbolicity of sliced spacetimes, and (3) computation of the causal boundary of a Cauchy temporal splitting. Based on arXiv:2110.13672.

Asymptotically hyperbolic (or Poincaré-Einstein) Einstein metrics are important in conformal geometry and the physics of the AdS-CFT correspondence. In 1991, Graham and Lee proved that any sufficiently small perturbation (in Holder norm) of the round metric on the unit sphere (with dimension at least 3) is the “conformal infinity” of an asymptotically hyperbolic Einstein metric on the open unit ball. In this talk I will give the background and review the proof of this result. I then discuss recent joint work with Rochon proving the existence of Einstein metrics near certain geometrically finite quotients of the hyperbolic metric.

Regge-Teitelboim conditions (and their relationship with harmonic coordinates)

In General Relativity, an “isolated system at a given instant of time” is modeled as an asymptotically Euclidean initial data set (M,g,K). Such asymptotically Euclidean initial data sets (M,g,K) are characterized by the existence of asymptotic coordinates in which the Riemannian metric g and second fundamental form K decay to the Euclidean metric \delta and to 0 suitably fast, respectively. Using harmonic coordinates Bartnik showed that (under suitable integrability conditions on their matter densities) the (ADM-)energy, (ADM-)linear momentum and (ADM-)mass of an asymptotically Euclidean initial data set are well-defined. To study the (ADM-)angular momentum and (BORT-)center of mass, however, one usually assumes the existence of Regge-Teitelboim coordinates on the initial data set (M,g,K) in question, i.e. the existence of asymptotically Euclidean coordinates satisfying additional decay assumptions on the odd part of g and the even part of K. We will show that, under certain circumstances, harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge-Teitelboim coordinates. This allows us to easily give examples of (vacuum) asymptotically Euclidean initial data sets which do not possess any Regge-Teitelboim coordinates. This is joint work with Carla Cederbaum and Jan Metzger.

A mainstream in mathematics is to study the relation between the geometry and topology of a manifold. The geometry is about the distance, length, area, volume and determined by its curvature. The topology is about properties that are preserved under continuous deformations. It is generally observed that a suitable assumption on the geometry would lead to certain restrictions on the topology. In this talk, we’ll discuss some recent progress in this direction, particularly the joint work with Xiaodong Cao and Matthew Gursky leading to a resolution of a conjecture proposed by S. Nishikawa.

Finsler spacetimes and its applications to cosmology and wildfire propagation

We will first show how Finsler spacetimes naturally appear as a tool to solve the time-dependent Zermelo problem in a manifold M, or more generally, the problem of finding the shortest trajectory in time when the velocity is prescribed at any direction and any instant of time, namely, the velocity is a function of the direction and the time. It turns out that the shortest trajectories are the projections to M of lightlike geodesics in the non-relativistic spacetime R x M, where the first coordinate is the absolute time. These findings can be applied to wildfire propagation models as the velocity of the fire in every direction and instant of time is prescribed, namely, it depends on the wind, the slope, the vegetation, humidity… so the propagation of the fire can be obtained computing the orthogonal lightlike geodesics to the firefront. On the other hand, Finsler spacetimes can be used as cosmological models in situations with a certain degree of anisotropy. We will discuss the meaning of the stress-energy tensor in this context and some proposals for Einstein field equations.

Let (M^n, g) denote a smooth and compact Riemannian manifold with no boundary equipped with a smooth Riemannian metric g. Courant’s nodal domain theorem asserts that for the Laplace-Beltrami operator on M, if we order the eigenvalues in increasing order with multiplicity, then the eigenfunction for the k-th eigenvalue has at most k connected components(the nodal domains) where the eigenfunction does not vanish. C. Fefferman and H. Donnelley proposed about 30 years ago a local version of this result on every ball in M. This local question is connected with the question of S.-T. Yau on the length of the zero set of the eigenfunctions. We propose to give answers to this question. This work is joint with A. Logunov E. Mallinikova and D. Mangoubi.

The so called Yamabe problem in Conformal Geometry consists in finding a metric conformal to a given one and which has constant scalar curvature. From the analytic point of view, this problem becomes a semilinear elliptic PDE with critical (for the Sobolev embedding) power non-linearity. If we study the problem in the Euclidean space, allowing the presence of nonzero-dimensional singularities can be transformed into reducing the non-linearity to a Sobolev-subcritical power. A quite recent notion of non-local curvature gives rise to a parallel study which weakens the geometric assumptions giving rise to a non-local semilinear elliptic PDE.

In this talk, we will focus on metrics which are singular along nonzero-dimensional singularities. In collaboration with Ao, Chan, Fontelos, González and Wei, we covered the construction of solutions which are singular along (zero and positive dimensional) smooth submanifolds in this fractional setting. This was done through the development of new methods coming from conformal geometry and Scattering theory for the study of non-local ODEs. Due to the limitations of the techniques we used, the particular case of “maximal” dimension for the singularity was not covered. In a recent work, in collaboration with H. Chan, we cover this specific dimension constructing and studying singular solutions of critical dimension.

I will discuss two new results in rigidity Kähler geometry. I will first talk about the (almost) rigidity of Kähler manifolds with positive Ricci curvature and almost maximal volume. I will then discuss the rigidity of Kähler manifolds with positive bi-sectional curvature and maximal diameter. The first result is a complex analogue of a famous volume rigidity theorem of Colding, and the second result is of course the complex analogue of the well-known maximal diameter theorem of Cheng. This is joint work with Harish Seshadri and Jian Song.

Intrinsic flat stability of the positive mass theorem for graphical manifolds

The rigidity of the Riemannian positive mass theorem for asymptotically flat manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the Euclidean space. This leads us to ask us if an asymptotically flat manifold that has total mass almost zero is close, in some sense, to the Euclidean space. I will review Huang-Lee-Sormani, Allen-Perales and Huang-Lee-Perales’s PMT stability result for asymptotically flat graphical manifolds where intrinsic flat distance was used. Motivated by these, and as a part of an ongoing project with A. Cabrera Pacheco and M. Graf, I will also discuss an analogous result for asymptotically hyperbolic graphical manifolds.

Motivated by Morse-theoretic considerations, Yau asked in 1987 whether all minimal 2-spheres in a 3-dimensional ellipsoid inside R^4 are planar, i.e., determined by the intersection with a hyperplane. Recently, this was shown not to be the case by Haslhofer and Ketover, who produced an embedded non-planar minimal 2-sphere in sufficiently elongated ellipsoids, combining Mean Curvature Flow and Min-Max methods. Using Bifurcation Theory and the symmetries that arise if at least two semiaxes coincide, we show the existence of arbitrarily many distinct embedded non-planar minimal 2-spheres in sufficiently elongated ellipsoids of revolution. This is based on joint work with P. Piccione.

Geodesics on SL(n) with the Hilbert-Schmidt metric, and one application to fluids

One way to analyze the motion of rigid bodies is to study the corresponding trajectory in SO(n); the physical features of the body give rise to a Riemannian metric. If we were to replace rigid bodies with incompressible fluids, the same procedure leads to Arnol’d’s equations on the volume-preserving diffeomorphism group, into which SL(n) sits as a subgroup. For some fluid bodies, this subgroup is totally geodesic; in this case the induced Riemannian metric on SL(n) is the Hilbert-Schmidt metric. Extending previous works by Sideris and Roberts-Shkoller-Sideris, we analyze the geodesic motion in this setting. Results are obtained in collaboration with my REU students Audrey Rosevear (Amherst) and Samuel Sottile (MSU).

Koebe circle domain conjecture and the Weyl problem in hyperbolic 3-space

In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl type problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo.

A positive-definite energy functional for the axisymmetric perturbations of Kerr-Newman black hole spacetimes.

The mathematical problem of stability of black hole solutions of Einstein’s equations is important to establish the astrophysical significance of these solutions. The problem concerns the stability of the Kerr(-Newman) family of solutions of the Einstein(-Maxwell) equations. In terms of mathematical technique, an important obstacle is that the energy of waves propagating through such spacetimes is not necessarily positive-definite due to the existence of the ergoregion for non-zero angular momentum. In this talk, we shall discuss the proof that there exists an energy functional for axially symmetric linear perturbations of Kerr-Newman that is positive-definite and strictly conserved. The proof is based on a Hamiltonian approach to the Einstein equations and holds for the full sub-extremal range. This is joint work with Vincent Moncrief.

Consider a small spherical tube around a compact submanifold M in Euclidean space. In 1939 Weyl showed that the volume of such a tube only depends on the radius of the tube and the intrinsic curvature of M. What happens for tubes with more complicated cross sections D? We will see that under sufficiently strong symmetry assumptions on D the tube volume turns out to be still intrinsic. Joint work with Gert Heckman.

Comparison results for three-dimensional manifolds.

Typical comparison results in geometry require information on Ricci curvature. In dimension three, we show that this can sometimes be relaxed to scalar curvature. An example is a sharp upper bound for the bottom spectrum of the Laplacian on noncompact three-dimensional manifolds.

Manifold submetries are a geometric generalization of isometric group actions, as well as of singular Riemannian foliations. For the round sphere, we show that manifold submetries are in one-one correspondence with a certain class of algebras of polynomials intimately related to the Laplace operator. Surprisingly, this broader point of view yields interesting new results in Classical Invariant Theory. (Joint work with Marco Radeschi)

We will consider the problem of constructing singular metrics of constant non-local curvature in conformal geometry, using a gluing scheme. This non-local curvature is defined from the conformal fractional Laplacian, a Paneitz type operator of non-integer order. For the gluing process, one needs a model solution which is given by the solution of a non-local ODE with good conformal properties. It turns out that conformal geometry provides powerful tools for the analysis of such equations.

Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations.

In this talk I’ll discuss structure results for homogeneous Riemannian spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function. Our results generalize earlier rigidity results for gradient Ricci solitons and warped product Einstein metrics. In particular, they provide results for homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics. This is joint work with P. Petersen of UCLA.

Classification of Ground States for Critical Dirac Equations.

In this talk I will present a classification result for nonlinear Dirac equations with critical nonlinearities on the Euclidean space. They appear naturally in conformal spin geometry and in variational problems related to critical Dirac equations on spin manifolds. Moreover, two-dimensional critical Dirac equations recently attracted a considerable attention as effective equations for wave propagation in honeycomb structures. Exploiting the conformal invariance of the problem ground state solutions can be classified, in analogy with the well-known result for the Yamabe equation. This is a joint work with Andrea Malchiodi (SNS, Pisa) and Ruijun Wu (SISSA, Trieste).

Conformally Kähler Einstein-Maxwell metrics on blow-ups.

Conformally Kähler Hermitian metrics of constant Riemannian scalar curvature and J-invariant Ricci are called conformally Kähler Einstein-Maxwell metrics. In this talk, we discuss deformations and possible constructions of such metrics on blow-ups. This is a joint work in progress with Abdellah Lahdili.