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Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims that any conf …

$\DeclareMathOperator\ddiv{div}\DeclareMathOperator\tr{tr}\newcommand{\conf}{\mathrm{conf}}$Consider this PDE on a symmetric tensor $h$ on $S^2$: $$\Delta \text{tr}(h) - \ddiv(\ddiv(h)) + \tr(h) = f$$ …

Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying: $$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$ Where $(r,\theta,\phi)$ …

Let $\mathfrak{X}_{CK}^{\perp}$ be the space of vector fields on $S^2$ that are $L^2$-orthogonal to conformal Killing vector fields. Let $\mathfrak{X}_{CK}$ be the 6-dimensional space of conformal Kil …

Let $\gamma$ be a metric on $S^2$. I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$: $$div_{\gamma} A = \omega$$ where $\omega$ is a 1-form. It is known that there ex …

Consider the manifold $M := \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $\gamma$ be the induced metric on $\partial M$ …

Let $M = [1,\infty) \times S^2$. Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies $$h = O(1/r),\quad \part …

Consider a family of metrics and functions $(g(t), u(t))$ on $M:= \mathbb{R}^3 \setminus B_1$ satisfying $$ g(0) = g_0, \quad g'(0) = \tilde g, \quad u(0) = u_0, \quad u'(0) = \tilde u$$ where $g_0$, …

Let $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball (equip $M$ with the euclidean metric for simplicity, but it will be replaced by an arbitrary asymptotically flat metric). Let $N: L^2( …

Consider the space $\mathcal{A}$ of functions $\Omega$ such that $\Omega^2 \gamma_0$ is isometric to the round sphere, where $\gamma_0$ is the round sphere. (so $\Omega^2 \gamma_0$ is of constant curv …

Let $M = [1,\infty)\times S^2$. Let $\Omega$ be any smooth positive function on $S^2$. Is the metric $dr^2+ \Omega^2 r^2 g_\text{round}$ asymptotically flat (where $g_\text{round}$ is the round metric …

Consider the PDE $$\Delta f + \lambda f = g$$ on $S^2$, where $\Delta$ is with respect to the round metric, $g \in L^2(S^2)$ and $\lambda>0$. I wish to study the existence and uniqueness of this PDE f …

Let $M = \mathbb{R}^3 \setminus \overline{B_1}$ where $\overline{B_1}$ is the closed unit ball. Let $g$ be an asymptotically flat metric of the form $g_{ij} = \delta_{ij}+h_{ij}$ in standard coordinat …

Let $(r,\theta,\phi)$ be the spherical coordinates on $\mathbb{R}^3$ where $\theta \in (0,\pi)$ and $\phi\in (0,2\pi)$. Does there exist an asymptotically flat metric $g$ on $\mathbb{R}^3\setminus B_1 …

Suppose we have the manifold $\mathbb{R}^3$ equipped with a Riemanian metric $g$ (not necessarily the Euclidean metric. And the induced metric on the $B_1$ (the ball with radius $1$) is $\gamma$. Su …