北京雁栖湖应用数学研究院

In $n$-spacetime dimensions, the Einstein--scalar field equations with an exponential potential are given by $$R_{ij} = 2 \nabla_i \phi \nabla_j \phi + \frac{4}{n-2} V(\phi) g_{ij},$$ $$\Box_{g} \phi = V'(\phi),$$where $V(\phi) = V_0 e^{-s\phi}$, $V_0 \in \{-1,0,1\}$ and $s\in \mathbb{R}$. The Kasner--scalar field spacetimes are a distinguished family of solutions to the Einstein-scalar field equations that are past geodesically incomplete. In these solutions, past-directed timelike geodesics terminate at a spacelike big bang singularity characterised by curvature blow-up. Remarkable progress has been made recently on establishing the past stability of these solutions and their big bang singularities. The first major breakthrough was achieved by Fournodavlos-Rodnianski-Speck, who proved stability over the full sub-critical range of Kasner exponents in the case of a vanishing potential, i.e.~$V_0 = 0$. Subsequently, Oude Groeniger-Petersen-Ringstr\"{o}m established past stability for the Kasner-scalar field spacetimes with non-vanishing potentials, $|V_0| \neq 0$, under the condition $s < \sqrt{\frac{8(n-1)}{n-2}}$. In both settings, perturbations of these spacetimes terminate in the past at quiescent, generically anisotropic big bang singularities that are characterised by curvature blow-up. For the parameter choices $V_0 = -1$ and $s > \sqrt{\frac{8(n-1)}{n-2}}$, the Einstein-scalar field equations admit a distinct family of isotropic solutions with big bang singularities, known as \textit{ekpyrotic FLRW spacetimes}. In this talk, I will present a new proof of big bang stability for this family. A remarkable feature of perturbations of these solutions is that, unlike perturbations of the Kasner-scalar field family, anisotropies are dynamically suppressed, and the spacetimes isotropise as they approach quiescent, spacelike big bang singularities characterised by curvature blow-up.

We characterize spin initial data sets that saturate the BPS bound in asymptotically AdS spacetimes. Our results show that (1) null imaginary Killing spinors give rise to codimension-2 foliations corresponding to Siklos waves, and (2) any imaginary Killing spinor of mixed causal type can be reduced to a strictly timelike or null spinor. This is joint work with Sven Hirsch.

The question of singularity formation in fluid dynamics remains one of the most challenging open problems in mathematical physics. In this talk, we present new results showing that solutions to 3D hypodissipative Navier-Stokes equations with smooth initial data and an external forcing that is integrable in C1+ϵ can break down in finite time. The dissipation in the equation amounts to 0.1 orders of derivative, or (-∆)^0.05. Time permitting, I will discuss extensions that allow for more dissipation and rougher forcing.

We prove the local existence theorem and establish an extension principle for the spherically symmetric Einstein Yang--Mills system (SSEYM) with $H^1$ data, which further implies Cauchy stability for the system. Based on this result, we further prove an extension theorem for developments of weighted $H^1$ data. The weighted $H^1$ space allows H\{o}lder continuous data. In contrast to a massless scalar field, the purely magnetic Yang--Mills field in spherical symmetry satisfies a wave-type equation with a singular potential. As a consequence, the proof of Christodoulou, based on an $L^\infty-L^\infty$ estimate, fails in the Yang--Mills context. Instead, we employ an $L^2$-based method, which is valid for both massless and massive scalar matter fields as well. These are based on joint works with Junbin Li.

We construct asymptotically flat vacuum initial data without trapped surfaces whose Einstein evolution leads to the formation of several disjoint trapped regions in finite time. The construction combines Christodoulou’s short pulse method with a localized gluing procedure for the Einstein constraint equations, in which neighborhoods of the poles of a Brill-Lindquist manifold are replaced by constant-time slices of suitable dynamical spacetimes, while the data remain exactly Brill-Lindquist outside. We will also discuss some follow-up works. This is based on joint work with Elena Giorgi and Dawei Shen (Columbia University).

In this talk, I will discuss some positive mass theorems for certain singular spaces inspired by the dimension reduction techniques employed in the study of the geometry of manifolds with positive scalar curvature. In these results, we assume only that the scalar curvature is non-negative in a strong spectral sense, which aligns well with the stability condition of a minimal hypersurface in an ambient manifold with non-negative scalar curvature. As an application, we provide a characterization of asymptotically flat (AF) manifolds with arbitrary ends, non-negative scalar curvature, and dimension less than or equal to 8.This also leads to positive mass theorems for AF manifolds with arbitrary ends and dimension less than or equal to 8 without using N.Smale's regularity theorem for minimal hypersurfaces in a compact 8-dimensional manifold with generic metrics. The talk is based on my recent joint work with He Shihang and Yu Haobin.

Localized deformations and gluing constructions for initial data sets are fundamental tools in general relativity. For interior domains, this field was pioneered by Corvino, who established the local surjectivity of the scalar curvature operator. This work was later extended to the full constraint map by Corvino-Schoen, and developed into a systematic theory using weighted spaces by Chruściel–Delay, Carlotto–Schoen, Corvino–Huang, and others. In this talk, I will discuss how these results can be generalized to the boundary setting under generic conditions, highlighting the unique analytical challenges that arise in this context. We will also examine the non-generic case, where various geometric constraints emerge, and discuss the resulting rigidity theorems and their connections to the positive mass theorem.

The Kerr de Sitter geometry models a rotating black hole in an expanding universe. I will review its stability properties in the context of the Einstein vacuum equations with positive cosmological constant, and present a recent resolution of the non-linear stability problem for the cosmological region. The talk is based on joint work with G Fournodavlos, and describes among others contributions by H Friedrich, P Hintz and A Vasy.

The dynamics of inviscid, compressible fluids, whether relativistic or classical, are described by hyperbolic systems. If nonlinear, these systems are prone to singularity formation, meaning that even small data with high regularity can launch solutions that blow up in finite time. As it turns out, such behavior is generic for conservation laws in low dimension. However, with an appropriate damping source, the formation of singularities like shocks is suppressed for sufficiently small data. In this talk we will discuss how expanding spacetimes, as found in cosmology, can create such a damping effect and explore the dynamics of these competing mechanisms in various settings.

We study the two-dimensional acoustical rarefaction waves under the irrotational assumptions. We provide a new energy estimates without loss of derivatives. We also give a detailed geometric description of the rarefaction wave fronts. As an application, we show that the Riemann problem is structurally stable in the regime of two families of rarefaction waves. This is a joint work with Prof. Pin Yu in Tsinghua University.

The positive energy theorems are a fundamental pillar in mathematical general relativity. Originally proved by Schoen-Yau and later Witten, these theorems were established for asymptotically flat manifolds where the metric tends to the standard Euclidean metric and whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is motivated by the desire to model an isolated gravitational system with a Minkowski space background for the spacetime. However, actual astrophysical massive objects are not truly isolated but rather exist within an expanding cosmological universe, where the second fundamental form is umbilic. With this in mind, we seek a notion of energy for initial data sets with an umbilic second fundamental form. In this talk, I present a definition of energy in such an expanding cosmological setting. Instead of Minkowski space, we take de Sitter space as the background spacetime, which, when written in flat-expanding coordinates, is foliated by umbilic hypersurfaces each isometric to Euclidean 3-space. This cosmological setting necessitates a quasi-local energy definition, as the presence of a cosmological horizon in de Sitter space obstructs a global one. We define energy in this quasi-local setting by adapting the Liu-Yau energy to our framework and establish positivity of this energy for certain bounded values of the cosmological constant. This is joint work with Annachiara Piubello and Rodrigo Avalos.

We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on 2-spheres, which we solve by an iteration procedure. Our results provide a large class of exterior solutions of the constraint equations that can be matched to given interior solutions, according to the existing gluing techniques. In particular, our result can be applied to provide a large class of initial Cauchy data sets evolving to black holes, generalizing the well-known result of the formation of trapped surfaces due to Li–Yu.

In general relativity it is of great interest to construct spacetimes satisfying the vacuum Einstein equations. While the Cauchy problem for vacuum Einstein equations has been well studied since the work of Choquet-Bruhat, the initial boundary value problem (IBVP) remains much less understood. To establish a well-posed IBVP, one needs to impose boundary conditions on the time-like boundary which give rise to both energy estimate and geometric uniqueness. Due to complexity of the problem, so far there has not been a canonical choice of boundary conditions. In this talk I will discuss properties of various choices of geometric boundary conditions for the IBVP based on a series of works joint with Michael Anderson.