Related papers: Hawking's singularity theorem for $C^{1,1}$-metric…

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

Following Weaver we study generalized differential operators, called (metric) derivations, and their linear algebraic properties. In particular, for k = 1, 2 we show that measures on k-dimensional Euclidean space that induce rank-k modules…

The Belinkskii, Khalatnikov and Lifshitz conjecture says that as one approaches space-like singularities in general relativity, 'time derivatives dominate over spatial derivatives' so that the dynamics at any spatial point is well captured…

We consider the Hawking-Penrose singularity theorems and the Lorentzian splitting theorem under the weaker curvature condition of nonnegative Bakry-Emery-Ricci curvature $Ric_f^m$ in timelike directions. We prove that they still hold when…

The original singularity theorems of Penrose and Hawking were proved for matter obeying the Null Energy Condition or Strong Energy Condition respectively. Various authors have proved versions of these results under weakened hypotheses, by…

We give a new self-contained proof of Bronshtein's theorem, that any continuous root of a $C^{n-1,1}$-family of monic hyperbolic polynomials of degree $n$ is locally Lipschitz, and obtain explicit bounds for the Lipschitz constant of the…

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $\mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that…

Given a function $f : A \to \mathbb{R}^n$ of a certain regularity defined on some open subset $A \subseteq \mathbb{R}^m$, it is a classical problem of analysis to investigate whether the function can be extended to all of $\mathbb{R}^m$ in…

Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a K\"ahler metric $\omega$ and a closed $(1, 1)$-form $\alpha$. Assuming a uniform estimate for…

In this paper, we obtain general conditions under which the wave equation is well-posed in spacetimes with metrics of Lipschitz regularity. In particular, the results can be applied to spacetimes where there is a loss of regularity on a…

We investigate the behavior of the solutions of a class of certain strictly hyperbolic equations defined on $(0,T]\times \mathbb{R}^n$ in relation to a class of metrics on the phase space. In particular, we study the global regularity and…

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $\Sigma$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation…

We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with…

In the paper, we prove the existence and uniqueness results of the $C^{1,1}$ regular graphic hypersurface for Dirichlet problem of a class of degenerate Hessian quotient type curvature equations under the condition $\psi^{\frac{1}{k-l}}\in…

We present a proof of the existence of the Hawking radiation for massive bosons in the Schwarzchild-de Sitter metric. It provides estimates for the rates of decay of the initial quantum state to the Hawking thermal state. The arguments in…

We prove that causal maximizers in $C^{0,1}$ spacetimes are either timelike or null. This question was posed in [17] since bubbling regions in $C^{0,\alpha}$ spacetimes ($\alpha <1$) can produce causal maximizers that contain a segment…

Locality plays a key role in the derivation of Hawking radiation. Could relaxing the locality criterion modify Hawking radiation? We consider Hawking radiation in a general class of non local quantum field theories. We prove that Hawking…

We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then…

We prove the existence of a $C^{1,1}$ conformally compact Einstein metric on the ball that has asymptotic sectional curvature decay to $-1$ plus terms of order $e^{-2r}$ where $r$ is the distance from any fixed compact set. This metric has…

We give a short and self-contained proof of the interior $\mathcal C^{1,1}$ regularity of solutions $\varphi:\Omega \to \mathbb{R}$ to the eikonal equation $|\nabla \varphi|=1$ in an open set $\Omega\subset \mathbb{R}^{N}$ in dimension…