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## 数学代写|黎曼几何代写Riemannian geometry代考|Hyperbolicity of Spacelike Hypersurfaces

In this section we shall study the hyperbolicity of spacelike hypersurfaces with controlled mean curvatures in spacetimes with timelike sectional curvatures bounded from below. For that purpose, and motivated by the Riemannian results above, we naturally apply the analysis of the Lorentzian distance function, which was already presented in Section 4 .

First of all, we recall (see Theorem 15.1), that a Riemannian manifold $M$ is hyperbolic (non-parabolic) if and only if there exists a non-constant subharmonic function which is bounded from above and globally defined on $M$.
Remark 20.1.
i) This definition is equivalent to the fact that there exists a non-constant positive superharmonic function globally defined on $M$. To see the equivalence, observe that if $f$ is a non-constant subharmonic function bounded from above on $M$, then choosing $C>\max _{M} f$ we obtain $C-f$ a non-constant positive superharmonic function. Conversely, if $f$ is a non-constant positive superharmonic function on $M$, then $-f$ is a non-constant subharmonic function bounded from above on $M$.
ii) On the other hand, the existence of a non-constant positive superharmonic function $f$ globally defined on $M$ is equivalent to the existence of a nonconstant bounded (from above and from below) subharmonic function globally defined on $M$.

To see the equivalence observe that if $f$ is a non-constant bounded (from above and from below) subharmonic function on $M$, then choosing $C>\max _{M} f$ we obtain $C-f$ a non-constant positive superharmonic function. Conversely, if $f$ is a non-constant positive superharmonic function on $M$, then $0<\frac{f}{\sqrt{1+f^{2}}} \leq 1$ determines a non-constant bounded (from above and from below) subharmonic function.

As a consequence of our previous results, (see Section 4), we have the following
Theorem 20.2 ([AHP]). Let $N^{n+1}$ be an $(n+1)$-dimensional spacetime, $n \geq 2$, such that $K_{N}(\Pi) \geq b$ for all timelike planes in $N$. Assume that there exists a point $p \in N^{n+1}$ such that $\mathcal{I}^{+}(p) \neq \emptyset$, and let $\psi: \Sigma \rightarrow N^{n+1}$ be a spacelike hypersurface with $\psi(\Sigma) \subset \mathcal{I}^{+}(p)$. Let us denote by u the function $d_{p}$ along the hypersurface, and assume that $u \leq \pi /(2 \sqrt{-b})$ if $b<0$. Then (i) If the future mean curvature of $\Sigma$ satisfies
$H \leq \frac{2 \sqrt{n-1}}{n} f_{b}(u) \quad\left(\right.$ with $H0$ and $H \leq \frac{2 \sqrt{n-1}}{n} \sqrt{b}$, then $\Sigma$ is hyperbolic.
In particular, every maximal hypersurface contained in $\mathcal{I}^{+}(p)$ (and satisfying $u<$ $(\pi / 2 \sqrt{-b})$ if $b<0)$ is hyperbolic.

## 数学代写|黎曼几何代写Riemannian geometry代考|Weighted Riemannian Manifolds

The notion of weighted manifolds generalizes the notion of Riemannian manifolds, so we will use this section for two purposes: Firstly to describe the natural questions that arise from considering the type problem, i.e., parabolicity versus hyperbolicity, in this new and wider context; secondly to give an account of some of the new results, (which includes the Riemannian cases) in this field from the last 20 years.
We shall present some of the main results obtained in the preprints [HPR1] and [HPR2] by C. Rosales and the first and third named authors, with due reference to the previous works and results concerning the weighted setting of several authors, [Ba, Mo, Gri2, Gri3, GriMa, W, Q, WW, PRRS], as well as to the work of the second and the third named authors in the last years, [MaP4, MaP5, MaP7, $\mathrm{EP}$, which are concerned with the Riemannian case and also continues the results presented in the preceding sections.

Let $(N, g)$ be a complete Riemannian manifold. A density $e^{h}$, where $h: N \rightarrow$ $\mathbb{R}$ a smooth function on $N$, is used to put a controlled weight on the Hausdorff measures associated to the Riemannian metric. In particular, for any Borel set $E \subseteq N$, and any $C^{1}$ hypersurface $P \subseteq N$, the weighted volume of $E$ and the weighted area of $P$ are given by
$$\operatorname{Vol}{h}(E):=\int{E} d V_{h}=\int_{E} e^{h} d V, \quad \operatorname{Vol}{h}(P):=\int{P} d A_{h}=\int_{P} e^{h} d A,$$
where $d V$ and $d A$ denote the Riemannian elements of volume and area, respectively, (see [Mo, Ch. 18] for an introduction to this generalization of Riemannian geometry).

The density function determines not only generalizations of volume and area, but also generalizations of some key differential operators on Riemannian manifolds. We will denote the background metric of the Riemannian manifold by $g=\langle., .\rangle$. We define the weighted Laplacian or $h$-Laplacian of a function $u \in C^{2}(N)$ as in [Gri2, Sect. 2.1],
$$\Delta_{h} u:=\Delta u+<\nabla h, \nabla u>,$$
where $\Delta$ and $\nabla$ stand for the Laplace-Beltrami operator and the gradient of a function, respectively.

Given a domain (connected open set) $\Omega$ in $N$, a function $u \in C^{2}(\Omega)$ is $h$ harmonic (resp. h-subharmonic) if $\Delta_{h} u=0$ (resp. $\Delta_{h} u \geq 0$ ) on $\Omega$. As in the unweighted setting there is a strong maximum principle and a Hopf boundary point lemma for $h$-subharmonic functions, that reads exactly as Theorem 12.4, but replacing subharmonic functions by $h$-subharmonic functions.

Also, as in the unweighted context, from this weighted maximum principle it is clear that any $h$-subharmonic function on a compact manifold $N$ must be constant, and it is natural to wonder what happens in the non-compact case. This question leads to the notion of weighted parabolicity, (see Theorem $15.1$ for the Riemannian approach).

## 数学代写|黎曼几何代写Riemannian geometry代考|Weighted Capacities

The notion of weighted capacity $\operatorname{Cap}{h}(K)$ of a precompact set $K$ plays a key role in the study of the type problem concerning the parabolicity-versus-hyperbolicity question. In particular, the $h$-parabolicity of $N$ is again characterized by the fact that $\operatorname{Cap}{h}(K)=0$ for any/some compact set $K \subseteq M$ with non-empty interior (see [GriSa]). In this section we present the notion of weighted capacity, thereby revising and extending to the weighted context the notions presented in Section $14 .$
Next, we will recall how the $h$-parabolicity of manifolds can be characterized by means of weighted capacities. Let $\Omega \subseteq N$ be an open set and $K \subseteq \Omega$ a compact set. The weighted Newtonian capacity of the capacitor $(K, \Omega)$ is defined by
$$\operatorname{Cap}{h}(K, \Omega)=\inf {\phi \in \mathbb{L}(K, \Omega)} \int_{\Omega}|\nabla \phi|^{2} d V_{h},$$
where $\mathbb{L}(K, \Omega)$ denotes the set of locally Lipschitz functions on $M$ with compact support in $\Omega-\bar{K}$ which satisfy $0 \leq \phi \leq 1$ and $\phi_{\text {loк }}=0, \phi_{\text {|on }}=1$. Note that the extrinsic annulus $A_{\rho, R} \subseteq P \subseteq N$ in Section 9 corresponds to the capacitor $\left(D_{\rho}, D_{R}\right)$, while the capacitor $\left(B_{\rho}^{N}, B_{R}^{N}\right)$ is an intrinsic annulus. When $\Omega=N$, $\operatorname{Cap}{h}(K, N)=\operatorname{Cap}{h}(K)$ is the $h$-capacity of $K$ at infinity.

Moreover, if $\Omega \subseteq N$ is precompact, it can be proved, that the $h$-capacity of the compact set $K$ in $\Omega$ is given as the following integral, see [Gri1], [Gri2], and equation (14.5) in Section 14:
$$\operatorname{Cap}{h}(K, \Omega)=\int{\Omega}|\nabla \phi|^{2} e^{h} d V=\int_{\partial K}|\nabla \phi| e^{h} d A=\int_{\partial K} \frac{\partial u}{\partial \nu} d A_{h}$$
where the vector field $\nu$ is the outer unit normal along $\partial(\Omega \backslash K)$, i.e., the unit normal along $\partial K$ pointing into $K$, and $\phi$ is the solution of the Laplace equation on $\Omega-K$ with Dirichlet boundary values:
$$\left{\begin{array}{c} \Delta_{h} u=0, \ \left.u\right|{\partial K^{*}}=0, \ \left.u\right|{\partial \Omega}=1 . \end{array}\right.$$
In other words, the infimum in (22.1) is attained by the solution to the $h$-Laplace equation with Dirichlet condition on the boundary (22.3). The function $\phi$ is called the $h$-capacity potential of the capacitor $(K, \Omega)$.

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