### 数学代写|黎曼几何代写Riemannian geometry代考|MATH3342

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## 数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Lorentzian Distance Functions

For comparison, and before going further into the Riemannian setting, we briefly present the corresponding Hessian analysis of the distance function from a point in a Lorentzian manifold and its restriction to a spacelike hypersurface. The results can be found in [AHP], where the corresponding Hessian analysis was also carried out, i.e., the analysis of the Lorentzian distance from an achronal spacelike hypersurface in the style of Proposition 3.9. Recall that in Section 3 we also considered

the analysis of the distance from a totally geodesic hypersurface $P$ in the ambient Riemannian manifold $N$.

Let $\left(N^{n+1}, g\right)$ denote an $(n+1)$-dimensional spacetime, that is, a timeoriented Lorentzian manifold of dimension $n+1 \geq 2$. The metric tensor $g$ has index 1 in this case, and, as we did in the Riemannian context, we shall denote it alternatively as $g=\langle,$,$rangle (see, e.g., [O’N] as a standard reference for this section).$
Given $p, q$ two points in $N$, one says that $q$ is in the chronological future of $p$, written $p \ll q$, if there exists a future-directed timelike curve from $p$ to $q$. Similarly, $q$ is in the causal future of $p$, written $p<q$, if there exists a future-directed causal (i.e., nonspacelike) curve from $p$ to $q$.
Then the chronological future $I^{+}(p)$ of a point $p \in N$ is defined as
$$I^{+}(p)={q \in N: p \ll q} .$$
The Lorentzian distance function $d: N \times N \rightarrow[0,+\infty]$ for an arbitrary spacetime may fail to be continuous in general, and may also fail to be finite-valued. But there are geometric restrictions that guarantee a good behavior of $d$. For example, globally hyperbolic spacetimes turn out to be the natural class of spacetimes for which the Lorentzian distance function is finite-valued and continuous.

Given a point $p \in N$, one can define the Lorentzian distance function $d_{p}$ :
$M \rightarrow[0,+\infty]$ with respect to $p$ by
$$d_{p}(q)=d(p, q) .$$
In order to guarantee the smoothness of $d_{p}$, we need to restrict this function on certain special subsets of $N$. Let $\left.T_{-1} N\right|{p}$ be the following set $$\left.T{-1} N\right|{p}=\left{v \in T{p} N: v \text { is a future-directed timelike unit vector }\right} .$$
Define the function $s_{p}:\left.T_{-1} N\right|{p} \rightarrow[0,+\infty]$ by $$s{p}(v)=\sup \left{t \geq 0: d_{p}\left(\gamma_{v}(t)\right)=t\right},$$
where $\gamma_{v}:[0, a) \rightarrow N$ is the future inextendible geodesic starting at $p$ with initial velocity $v$. Then we define
$$\tilde{\mathcal{I}}^{+}(p)=\left{t v: \text { for all }\left.v \in T_{-1} N\right|{p} \text { and } 0{p}(v)\right}$$
and consider the subset $\mathcal{I}^{+}(p) \subset N$ given by
$$\mathcal{I}^{+}(p)=\exp {p}\left(\operatorname{int}\left(\tilde{\mathcal{I}}^{+}(p)\right)\right) \subset I^{+}(p) .$$ Observe that the exponential map $$\exp {p}: \operatorname{int}\left(\tilde{\mathcal{I}}^{+}(p)\right) \rightarrow \mathcal{I}^{+}(p)$$
is a diffeomorphism and $\mathcal{I}^{+}(p)$ is an open subset (possible empty).
Remark 4.1. When $b \geq 0$, the Lorentzian space form of constant sectional curvature $b$, which we denote as $N_{b}^{n+1}$, is globally hyperbolic and geodesically complete, and every future directed timelike unit geodesic $\gamma_{b}$ in $N_{b}^{n+1}$ realizes the Lorentzian distance between its points. In particular, if $b \geq 0$ then $\mathcal{I}^{+}(p)=I^{+}(p)$ for every point $p \in N_{b}^{n+1}$ (see [EGK, Remark 3.2]).

## 数学代写|黎曼几何代写Riemannian geometry代考|Concerning the Riemannian Setting and Notation

Returning now to the Riemannian case: Although we indeed do have the possibility of considering 4 basically different settings determined by the choice of $p$ or $V$ as the ‘base’ of our normal domain and the choice of $K_{N} \leq b$ or $K_{N} \geq b$ as the curvature assumption for the ambient space $N$, we will, however, mainly consider the ‘first’ of these. Specifically we will (unless otherwise explicitly stated) apply the following assumptions and denotations:
Definition 5.1. A standard situation encompasses the following:
(1) $P^{m}$ denotes an $m$-dimensional complete minimally immersed submanifold of the Riemannian manifold $N^{n}$. We always assume that $P$ has dimension $m \geq 2 .$
(2) The sectional curvatures of $N$ are assumed to satisfy $K_{N} \leq b, b \in \mathbb{R}$, cf. Proposition $3.10$, equation (3.13).
(3) The intersection of $P$ with a regular ball $B_{R}(p)$ centered at $p \in P$ (cf. Definition 3.4) is denoted by
$$D_{R}=D_{R}(p)=P^{m} \cap B_{R}(p)$$
and this is called a minimal extrinsic $R$-ball of $P$ in $N$, see the Figures 3-7 of extrinsic balls, which are cut out from some of the well-known minimal surfaces in $\mathbb{R}^{3}$.
(4) The totally geodesic $m$-dimensional regular $R$-ball centered at $\tilde{p}$ in $\mathbb{K}^{n}(b)$ is denoted by
$$B_{R}^{b, m}=B_{R}^{b, m}(\tilde{p})$$
whose boundary is the $(m-1)$-dimensional sphere
$$\partial B_{R}^{b, m}=S_{R}^{b, m-1}$$
(5) For any given smooth function $F$ of one real variable we denote
$$W_{F}(r)=F^{\prime \prime}(r)-F^{\prime}(r) h_{b}(r) \text { for } 0 \leq r \leq R$$
We may now collect the basic inequalities from our previous analysis as follows.

## 数学代写|黎曼几何代写Riemannian geometry代考|Green’s Formulae and the Co-area Formula

Now we recall the coarea formula. We follow the lines of [Sa] Chapter II, Section 5. Let $(M, g)$ denote a Riemannian manifold and $\Omega$ a precompact domain in $M$. Let $\psi: \Omega \rightarrow \mathbb{R}$ be a smooth function such that $\psi(\Omega)=[a, b]$ with $a<b$. Denote by $\Omega_{0}$ the set of critical points of $\psi$. By Sard’s theorem, the set of critical values $S_{\psi}=\psi\left(\Omega_{0}\right)$ has null measure, and the set of regular values $R_{\psi}=[a, b]-S_{\psi}$ is open. In particular, for any $t \in R_{\psi}=[a, b]-S_{\psi}$, the set $\Gamma(t):=\psi^{-1}(t)$ is a smooth embedded hypersurface in $\Omega$ with $\partial \Gamma(t)=\emptyset$. Since $\Gamma(t) \subseteq \Omega-\Omega_{0}$ then $\nabla \psi$ does not vanish along $\Gamma(t)$; indeed, a unit normal along $\Gamma(t)$ is given by $\nabla \psi /|\nabla \psi|$.
Now we let
\begin{aligned} &A(t)=\operatorname{Vol}(\Gamma(t)) \ &\Omega(t)={x \in \bar{\Omega} \mid \psi(x)<t} \ &V(t)=\operatorname{Vol}(\Omega(t)) \end{aligned}
Theorem 6.1.
i) For every integrable function $u$ on $\bar{\Omega}$ :
$$\int_{\Omega} u \cdot|\nabla \psi| d V=\int_{a}^{b}\left(\int_{\Gamma(t)} u d A_{t}\right) d t,$$
where $d A_{t}$ is the Riemannian volume element defined from the induced metric $g_{t}$ on $\Gamma(t)$ from $g$.
ii) The function $V(t)$ is a smooth function on the regular values of $\psi$ given by:
$$V(t)=\operatorname{Vol}\left(\Omega_{0} \cap \Omega(t)\right)+\int_{a}^{t}\left(\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{t}\right)$$
and its derivative is
$$\frac{d}{d t} V(t)=\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{t}$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Lorentzian Distance Functions

dp(q)=d(p,q).

\left.T{-1} N\right|{p}=\left{v \in T{p} N: v \text { 是一个面向未来的类时单位向量 }\right} 。\left.T{-1} N\right|{p}=\left{v \in T{p} N: v \text { 是一个面向未来的类时单位向量 }\right} 。

s{p}(v)=\sup \left{t \geq 0: d_{p}\left(\gamma_{v}(t)\right)=t\right},s{p}(v)=\sup \left{t \geq 0: d_{p}\left(\gamma_{v}(t)\right)=t\right},

\tilde{\mathcal{I}}^{+}(p)=\left{t v: \text { for all }\left.v \in T_{-1} N\right|{p} \text { 和} 0{p}(v)\right}\tilde{\mathcal{I}}^{+}(p)=\left{t v: \text { for all }\left.v \in T_{-1} N\right|{p} \text { 和} 0{p}(v)\right}

## 数学代写|黎曼几何代写Riemannian geometry代考|Concerning the Riemannian Setting and Notation

(1)磷米表示一个米黎曼流形的一维完全最小浸没子流形ñn. 我们总是假设磷有维度米≥2.
(2) 截面曲率ñ假设满足ķñ≤b,b∈R，参见。主张3.10，等式（3.13）。
(3) 交集磷用普通球乙R(p)以p∈磷（参见定义 3.4）表示为

DR=DR(p)=磷米∩乙R(p)

(4) 完全测地线米维规则R- 球为中心p~在ķn(b)表示为

∂乙Rb,米=小号Rb,米−1
(5) 对于任何给定的平滑函数F我们表示的一个实变量

## 数学代写|黎曼几何代写Riemannian geometry代考|Green’s Formulae and the Co-area Formula

i) 对于每个可积函数在上Ω¯ :

∫Ω在⋅|∇ψ|d在=∫一个b(∫Γ(吨)在d一个吨)d吨,

ii) 功能在(吨)是一个关于正则值的平滑函数ψ给出：

dd吨在(吨)=∫Γ(吨)|∇ψ|−1d一个吨

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