Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $(+-\cdots-)$, i.e. the signature of $g$ is $2-d$) with Levi-Civita connection $\nabla$, and consider the Klein-Gordon equation on $(M,g)$: $$\tag{1}\label{e1}\Box_g u+m^2 u=0\ ,\quad\Box_g=g^{-1}\nabla^2\ .$$ We assume $(M,g)$ to be *globally hyperbolic*: there is $\tau\in\mathscr{C}^\infty(M,\mathbb{R})$ surjective such that $\mathrm{d}\tau$ is a future-directed, timelike covector field and $\Sigma_t=\tau^{-1}(t)$ is a *Cauchy hypersurface* for $(M,g)$ (i.e. $M=$ the Cauchy development of $\Sigma_t$ in $(M,g)$) for all $t\in\mathbb{R}$. We then say that $\tau$ is a *Cauchy time function* for $(M,g)$, and one can show that $M$ is then diffeomorphic to $\mathbb{R}\times\Sigma_t$ for every $t\in\mathbb{R}$ (it suffices to follow the maximal orbits of a future-directed timelike vector field $T$ - e.g. $T=g^\sharp(\mathrm{d}\tau)$ -, which must intersect $\Sigma_t$ for all $t\in\mathbb{R}$ since they are all Cauchy hypersurfaces by hypothesis). We can (and shall) use a Cauchy time function $\tau$ to provide us a global time coordinate.

If $(M,g)=(\mathbb{R}^d,\eta)$ is Minkowski space-time with $\tau=$ the standard (Cartesian) time coordinate $x^0$, the stationary phase method (among others) allows us to show that if $u$ is a smooth solution of (1) with initial data of compact support and $m\neq 0$, then $$\tag{2}\label{e2}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-1}{2}}\Big)\ .$$ If $m=0$, we have instead $$\tag{3}\label{e3}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-2}{2}}\Big)\ .$$

On the other hand, if $M=\mathbb{R}\times\mathbb{S}^{d-1}\ni(t,\theta)$ and $g=\mathrm{d}t\otimes\mathrm{d}t-h$, where $h$ is the standard round metric on $\mathbb{S}^{d-1}$ (i.e. $(M,g)=$ the so-called *Einstein cylinder* or *Einstein static universe*), then in both cases we have *no decay at all*. Intuitively, this is due to the fact that all light rays are trapped in a compact spatial region due to the simple fact that $(M,g)$ has *compact* Cauchy hypersurfaces. For cases in between (e.g. Schwarzschild and Kerr geometries, etc.) with *non-compact* Cauchy hypersurfaces (which is the only case I am interested in from now on), there are two possible (not completely independent) phenomena due to focusing of null geodesics which slow down global decay of solutions of \eqref{e1} as compared with \eqref{e2} and \eqref{e3}:

*Trapping.*This means that at least some null geodesics remain within a compact*spatial*region (often lower-dimensional). This is the case of static black hole space-times such as Schwarzschild - light rays with initial position at the radius $r=3M$ (in $d=4$) and purely angular initial velocity stay forever in the*photosphere*$r=3M$;*Caustics.*This happens whenever (say, future-directed) null geodesics starting at a point $p\in M$ go beyond the (future) null conjugate locus of $p$.

Typically, trapping has a stronger effect in reducing global decay than caustics. On the other hand, caustics are usually *stable* against small perturbations of $g$, whereas trapping need *not* be. A lot of work has been dedicated by several people (Alinhac, Baskin, Sogge, Tataru, etc.) to investigate decay of solutions of \eqref{e1} when $(M,g)$ is either free of 1. and 2. or has a particular kind of trapping.

Having set up our context and framework, I am now finally able to pose my question. Consider the Klein-Gordon equation \eqref{e1} in a globally hyperbolic $(M,g)$ endowed with a *steep* Cauchy time function $\tau$ (i.e. $g^{-1}(\mathrm{d}\tau,\mathrm{d}\tau)\geq 1$). Since the set of all globally hyperbolic smooth Lorentzian metrics on $M$ is Whitney-$\mathscr{C}^0$-open in the space of all smooth Lorentzian metrics on $M$, we are able to perform small perturbations $g'$ of $g$ in that topology while keeping the globally hyperbolic character of $g'$. Moreover, if the perturbation is suitably small in that topology, a steep Cauchy time function remains a Cauchy time function with respect to $g'$.

Question:Suppose that smooth solutions to \eqref{e1} in $(M,g)$ with compactly supported initial data in $\Sigma_0$ have a time decay rate of the form $$\|u\|_{L^\infty(\Sigma_t)}=O(f(|t|))\ ,\quad t>0$$ for some $\delta>0$, where $f:(0,+\infty)\to(0,+\infty)$ is a strictly decreasing function satisfying $\lim_{t\to+\infty}f(t)=0$. Assume that the Cauchy hypersurfaces of $(M,g)$ arenotcompact. For a "generic" small perturbation $g'$ within the above conditions, what is the time decay rate for solutions $u'$ of the same kind to $$\tag{1'}\label{e1a}\Box_{g'} u'+m^2 u'=0\ ,\quad\Box_{g'}=g'{}^{-1}\nabla'{}^2$$ (here $\nabla'$ is the Levi-Civita connection associated to $g'$)? Is it faster than $O(f(|t|))$ for $g'$ "generic" but $g$ not so?

In other words, which is the best "structurally stable" global time decay rate we can expect from smooth solutions of the Klein-Gordon and wave equation with compactly supported initial data if $(M,g)$ has non-compact Cauchy hypersurfaces (to prevent situations where we know for sure there is no decay at all)?

I expect that loss of decay due to trapping should "generically" disappear but loss of decay due to caustics (which is supposedly milder) should "generically" remain, but I am not sure if this intuition is correct.

(**Remark:** Müller and Sánchez (*Lorentzian Manifolds Isometrically Embeddable in $\mathbb{L}^N$*, Trans. Amer. Math. Soc. **363** (2011) 5367-5379, arXiv:0812.4439 [math.DG]) and Minguzzi (*On the Existence of Smooth Cauchy Steep Time Functions*, Class. Quantum Grav. **33** (2016) 115001, arXiv:1601.05932 [gr-qc]) have shown that any globally hyperbolic $(M,g)$ admits a steep Cauchy time function)