Machine-translated from Chinese.
引子
几何分析是一门利用分析方法讨论几何问题的学科.
比如我们会在流形上讨论类似如下的分析问题:
对于$\varphi\in C^\infty(M,g),$ 考虑
$$
\left\{
\begin{aligned}
&-\Delta_gu=0\\
&u|_{\partial B_R(p)}=\varphi
\end{aligned}
\right.
$$
解是否存在唯一. 答案一般是否定的,
这样的分析问题也会反映了流形$M$上的几何性质. 类似的问题还有,
对光滑向量场$X,$ 如下公式是否会有好的形式?
$$
\int_{B_R(p)}\operatorname{div}X
$$
测地球
对于完备(以后默认完备)的黎曼流形$(M,g),$ $p\in M,$
我们有指数映射$\exp_p:T_pM\rightarrow M$为光滑映射,
$\exp_ptv$随$t$变动为测地线. 我们给出如下的记号:
$$
\widehat{C}_p=\{v\in T_pM|d\exp_p \text{沿$tv|_{t\in[0,1]}$非退化}\},\quad C_p=\exp_p\widehat{C}_p;
$$
$$
\widehat{S}_p=\{v\in T_pM|\,\exists\,\varepsilon, \exp_ptv|_{[0,1+\varepsilon]} \text{为线段(最短测地线)}\},\quad S_p=\exp_p\widehat{S}_p.
$$
称$S_p$为线段域(segment domain), $d\exp_p$退化的地方称为共轭点.
$\exp_p:\widehat{S}_p\approx S_p$为微分同胚. (是否要去掉零测集?)
一些事实是:
(i) $\widehat{S}_p\subset \widehat{C}_p$;
(ii) $\widehat{S}_p$为星形区域;
(iii) $\operatorname{Cut}_p:=M\setminus S_p$称为割迹,
有$\operatorname{Vol}(\operatorname{Cut}_p)=0.$
给出一个新定义:
$$
\operatorname{Inj}(p):=\inf \{r|B_r^n(0)\subset \widehat{S}_p\},
$$
称为$p$点的单射半径, 其中$B_r^n(0)\subset T_pM\cong \mathbb{R}^n.$
$r<\operatorname{Inj}(p)$时$B_r^g(p)\approx B_r^n(0)$为微分同胚,
而$r\ge \operatorname{Inj}(p)$时一定不是.
考虑将一条法棍长度不等分的弯成两半, 以凸出的弯折点为$p,$
那么有半径$r$使得$\partial B_r^g(p)$为一个圆环$S^1$和一个顶点$q$.
那么假设开始的问题有一个解$u\in C^2(B_r^g(p)),$ 那么它是有界调和函数,
从而$q$为可去奇点, 且$u$是光滑的. 记$\Omega:=B_r^g(p)\cup \{q\},$
则$\partial \Omega=S^1,$ $u\in C^\infty(\Omega).$
从而若$\varphi|_{S^1}\equiv 1,$ 则$\Omega$上有$u\equiv 1.$
因此倘若一开始边界条件有$\varphi(q)=0,$ 则该问题无解.
割迹
我们可以将测地球的边界分为两部分:
$$
(\partial B_r^g(p)\cap S_p)\cup (\partial B_r^g(p)\cap \operatorname{Cut}_p).
$$
由于$\widehat{S}_p$上$\exp_p$为微分同胚, 前半部分是一个$n-1$维子流形,
坏的部分主要集中在割迹上. 由于割迹是零测的, 坏的部分并不是太多.
事实上, 对于割迹, 我们有进一步的分解:
$$
\operatorname{Cut}_p=\{\exp_p v|r(d\exp_{p,v})\le n-1\}\cup \{x| \text{存在两条以上线段连接$p,x$}\}
$$
将前半部分记为$Q_p,$ 后面的部分中恰存在两条的记为$N_p,$ 其余记为$L_p.$
那么我们有如下的定理:
定理 1. $N_p$为光滑$n-1$维子流形, 且$\dim(L_p\cup Q_p)\le n-2.$
注意上述定理仅对光滑流形成立.
文章最后更新于 2022-09-15 21:20:34
Introduction
Geometric analysis is a discipline that uses analytical methods to discuss geometric problems.
For example, we will discuss analysis problems similar to the following on manifolds:
For $\varphi\in C^\infty(M,g),$ consider
$$
\left\{
\begin{aligned}
&-\Delta_gu=0\\
&u|_{\partial B_R(p)}=\varphi
\end{aligned}
\right.
$$
Whether there is a unique solution. The answer is generally no,
Such analysis problems will also reflect the geometric properties on the manifold $M$. Similar problems include,
Does the following formula have a good form for a smooth vector field $X,$?
$$
\int_{B_R(p)}\operatorname{div}X
$$
Measuring the Earth
For the complete (default complete from now on) Riemannian manifold $(M,g),$ $p\in M,$
We have the exponential map $\exp_p:T_pM\rightarrow M$ as a smooth map,
$\exp_ptv$ changes to a geodesic with $t$. We give the following notation:
$$
\widehat{C}_p=\{v\in T_pM|d\exp_p \text{沿$tv|_{t\in[0,1]}$非退化}\},\quad C_p=\exp_p\widehat{C}_p;
$$
$$
\widehat{S}_p=\{v\in T_pM|\,\exists\,\varepsilon, \exp_ptv|_{[0,1+\varepsilon]} \text{为线段(最短测地线)}\},\quad S_p=\exp_p\widehat{S}_p.
$$
Call $S_p$ a segment domain, and the degenerate part of $d\exp_p$ is called conjugate point.
$\exp_p:\widehat{S}_p\approx S_p$ is diffeomorphism. (Do you want to remove the zero test set?)
Some facts are:
(i) $\widehat{S}_p\subset \widehat{C}_p$;
(ii) $\widehat{S}_p$ is a star-shaped area;
(iii) $\operatorname{Cut}_p:=M\setminus S_p$ is called Cut trace,
Yes$\operatorname{Vol}(\operatorname{Cut}_p)=0.$
Give a new definition:
$$
\operatorname{Inj}(p):=\inf \{r|B_r^n(0)\subset \widehat{S}_p\},
$$
called $p$ point single shot radius, where $B_r^n(0)\subset T_pM\cong \mathbb{R}^n.$
When $r<\operatorname{Inj}(p)$, $B_r^g(p)\approx B_r^n(0)$ is diffeomorphism,
But $r\ge \operatorname{Inj}(p)$ is definitely not the case.
Consider bending a baguette in half with unequal lengths, and the protruding bending point is $p,$
Then there is radius $r$ such that $\partial B_r^g(p)$ is a ring $S^1$ and a vertex $q$.
So assuming that the starting problem has a solution $u\in C^2(B_r^g(p)),$, then it is a bounded harmonic function,
Therefore $q$ can remove singular points, and $u$ is smooth. Denote $\Omega:=B_r^g(p)\cup \{q\},$
Then $\partial \Omega=S^1,$ $u\in C^\infty(\Omega).$
So if $\varphi|_{S^1}\equiv 1,$ then there is $u\equiv 1.$ on $\Omega$
Therefore, if the initial boundary condition is $\varphi(q)=0,$, the problem has no solution.
Cut trace
We can divide the boundary of the Earth into two parts:
$$
(\partial B_r^g(p)\cap S_p)\cup (\partial B_r^g(p)\cap \operatorname{Cut}_p).
$$
Since $\exp_p$ on $\widehat{S}_p$ is a diffeomorphism, the first half is a $n-1$-dimensional submanifold,
The bad parts are mainly concentrated on the cutting traces. Since the cutting traces are zero-measured, there are not too many bad parts.
In fact, we have further decomposition of the cut trace:
$$
\operatorname{Cut}_p=\{\exp_p v|r(d\exp_{p,v})\le n-1\}\cup \{x| \text{存在两条以上线段连接$p,x$}\}
$$
Record the first half as $Q_p,$, the following part if there are exactly two of them as $N_p,$, and the rest as $L_p.$
Then we have the following theorem:
Theorem 1. $N_p$ is a smooth $n-1$ dimensional submanifold, and $\dim(L_p\cup Q_p)\le n-2.$
Note that the above theorem is only true for smooth flows.
The article was last updated on 2022-09-15 21:20:34
