数理解析研究所講究録 1293 巻 2002 年 150-153

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Caloric morphisms on\with respect to radial metrics$\{0\}$$\mathbb{R}^{n}$

茨城大学・理学部

ここでは,$M=\mathbb{R}^{n}\backslash \{0\}(n\geqq 2)$

勝孝 (Katsunori Shimomura )下村

Faculty of ScienceIbaraki University

上に radial を Riemann 計量を入れた多様体上でcaloric morphism を考える.
関数とする.の各点$\rho(|x|)(dx_{1}^{2}+\cdots+dx_{n}^{2})$ を radial を Riemann 計量とする.の Riemann 計量に関する熱方程式$\rho$を$(0, \infty)$

上の正値$C^{\infty}-$$\mathbb{R}^{n}\backslash \{0\}$$x$$\mathbb{R}\cross M$

Huに対して, $ds^{2}=$内のある領域上でこ$= \frac{\partial u}{\partial t}-\Delta_{\rho}u=0$を満たす関数 u を caloric function と呼ぶ. 但しここで$\Delta_{\rho}$

は M のラプラシアン$\Delta_{\rho}u=\sum_{i=1}^{n}\frac{1}{\rho(|x|)^{n/2}}\frac{\partial}{\partial x_{i}}(\rho(|x|)^{n/2-1}\frac{\partial u}{\partial x_{i}})$$= \frac{1}{\rho(|x|)}\Delta_{x}u+\frac{n-2}{2}\frac{\rho’(|x|)}{\rho(|x|)^{2}}\sum_{i=1}^{n}\frac{x_{i}}{|x|}\frac{\partial u}{\partial x_{i}}$

である.
D を

$\mathbb{R}\cross M$内の領域とし,$f(t, x)=(f_{0}(t, x_{1},$\ldots ,$x_{n}), f_{1}(t, x_{1},$\ldots ,$x_{n}),$\ldots ,$f_{n}(t, x_{1},$\ldots ,$x_{n}))$を $D$ から$\mathbb{R}\cross M$

への $C^{\infty}-$ 写像,

を $D$ 上の正値

$C^{\otimes}-$$f(D)$ 上の任意の caloric function $u(\tau, y)$ に対して,caloric function になるとき,$(f, \varphi)$関数とする.

$\varphi(t, x)(u\circ f)(t, x)$

が $D$ 上のを caloric morphism と呼ぶ.

M は Riemann 多様体であるから, [5] の特徴付けにより, (f,であるための必要十分条件は, 以下の (E-1)
で与えられる.

$\varphi)$が caloric morphism$-(\mathrm{E}-4)$

,

(E-1)

$H\varphi=0$(E-2)$\frac{\partial f_{i}}{\partial t}=\Delta_{\rho}f_{i}+2g(\nabla_{\rho}\log\varphi, \nabla_{\rho}f_{i})-\frac{df_{0}}{dt}(t)[(\Delta_{\rho}x_{i})\circ f]$(E-3)$\nabla_{\rho}f_{0}=0$(E-4)$g( \nabla_{\rho}f_{i}, \nabla_{\rho}f_{j})=\delta_{ij}\frac{df_{0}}{dt}(t)(\frac{1}{\rho}\circ f)$

,,,

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{

$\underline{\mathrm{B}}\llcorner^{arrow\vee}arrow\vec{-}\mathrm{C}^{\backslash }\backslash \nabla_{\rho}l\mathrm{f}M\emptyset$

gradient$\nabla_{\rho}u=\frac{1}{\rho(|x|)}\sum_{i=1}^{n}\frac{\partial u}{\partial x_{i}}\frac{\partial}{\partial x_{i}}$$i,j=1,$$n-\mathrm{C}^{\backslash }\backslash h\text{る}$$\ldots,$$\check{-}\emptyset\#\not\simeq\ovalbox{\tt\small REJECT} l1\backslash lf\mathrm{E}_{\mathrm{i}}\mathrm{f}\mathrm{f}\mathrm{l}\vee\backslash ,$.
$\mathrm{L}^{\backslash }\mathrm{J}\mathrm{T}\vee \mathrm{G}l\mathrm{f},$$\mathbb{R}\cross \mathbb{R}^{n}\text{上}\emptyset$$f(t, x)=(- \frac{1}{t}, \frac{x}{t})$,Appell Km$\varphi(t, x)=\frac{1}{|t|^{n/2}}\exp(\frac{-|x|^{2}}{4t})$$\ovalbox{\tt\small REJECT} f.’[] \mathrm{f}B_{\backslash _{l\backslash \backslash }}^{\mathrm{r}5\text{を}\mathrm{f}\mathrm{f}\mathrm{i}\mathfrak{p}\mathrm{R}_{\mathrm{J}}\Xi l\cdot\backslash }\backslash \backslash \backslash \iota 5\iota^{\sim}.\Xi T_{\grave{\wedge}}^{\pi\Re}$$\mathcal{O})\mathrm{g}\llcorner \mathrm{g}\sigma)\Psi_{L}\mathrm{f}\mathrm{f}\mathrm{i}\text{と}\llcorner\tau,$$f\theta^{\backslash ^{\backslash }}\backslash \ovalbox{\tt\small REJECT} l\iota 5.\text{を}\backslash \Phi\acute{\not\subset},$$f(t, x)=(f_{0}(t), \nu(t)R(t)x),$$\ovalbox{\tt\small REJECT} f_{arrow}’ l\mathrm{f}f(t, x)=(f_{0}(t), \nu(t)R(t)\frac{x}{|x|^{2}})$$(\nu(t)>0, R(t)|\mathrm{g}- \mathrm{g}_{\hat{\mathrm{A}}\acute{\mathrm{f}}\vec{\mathrm{T}}F^{1}\mathrm{J})t\sim}\llcorner_{-}\sim f_{|\mathrm{r}\circ\tau \mathrm{v}\backslash \text{る}}$$\text{る}$.
Riemann$\equiv-+\mathrm{p}\frac{\mathrm{H}}{\ovalbox{\tt\small REJECT}}p\backslash \backslash \backslash$radialcaloric morphism$(a\in \mathbb{R}, R\mathrm{t}\mathrm{f}\mathrm{g}\llcorner\hat{\mathrm{x}}\acute{\mathrm{f}}\vec{\mathrm{T}}F^{1}\mathrm{J}, C>0)[] \mathrm{f}$$41\backslash$$\langle$$\mathrm{f}\emptyset\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}\emptyset\rho\emptyset\Psi\nearrow\nearrow \text{を}\ovalbox{\tt\small REJECT}\wedge^{\backslash ^{\backslash }}$$\text{で}\hslash \text{る}p_{1}\sigma_{\mathit{2},}\mathbb{R}^{n}\emptyset\fbox \mathrm{o}\mathrm{E}\overline{\mathrm{z}}R\sigma^{\backslash }\mathbb{R}\emptyset\backslash \mp’\acute{\mathrm{f}}\overline{\mathrm{T}}\mathscr{E}\Phi$$f(t, x)=(t+a, Rx)$ ,$k*\iota \mathrm{k}\backslash \mathrm{J}_{\backslash }Fk\#_{\sim}arrow,$$\text{と},$,$\varphi(t, x)=C$

caloric morphism$\equiv-b\square \dagger \mathrm{f}^{\backslash }\Xi \mathfrak{M}\text{で}\prime s\mathrm{v}\backslash b\mathrm{C}7)\ arrow.\ovalbox{\tt\small REJECT}$$l\mathrm{f}1\backslash \mathrm{J}_{\backslash }^{-}\mathrm{F}\emptyset \mathrm{b}\emptyset\theta^{\backslash }\backslash ^{\backslash }\hslash \text{る}$.

1. (Appell transformation) p $>0,$ k$\in \mathbb{R}$.
. Mn$t_{arrow\gamma_{\epsilon}}^{\wedge}c\text{る}$(k $\neq-2),$$\mathrm{b}$metric$\rho t^{-}arrow \mathrm{i}\mathrm{g}\mathrm{t}\backslash \ovalbox{\tt\small REJECT}\rfloor\beta \mathrm{f}\mathrm{l}\theta^{\backslash }\backslash \backslash$$\rho(r)=pr^{k}\text{と}T\text{る}$.

$\ll^{-\emptyset}$$6$$f(t, x)=( \frac{at+b}{ct+d}, \frac{R}{|ct+d|^{2/(k+2)}}x)$,$\varphi(t, x)--C\exp[\frac{-p|x|^{k+2}}{ct+d|^{n/2}(k+2)^{2}|ct+d|}]|$$(\llcorner \mathrm{B}\llcorner a, b, c, d\in \mathbb{R}\mathrm{s}.\mathrm{t}.

ad-bc=1, C>0, Rl\mathrm{f}\mathrm{B}\llcorner K\mathrm{h}^{-}F\mathrm{I}\mathrm{J})l\mathrm{f}$. $-x,$ $k=-2\sigma$$k=-2\emptyset\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}\#\mathrm{f},\vee.0\pi\acute{\nearrow},\emptyset\ovalbox{\tt\small REJECT} \mathrm{x}\mathrm{R}<\mathrm{t}\mathrm{f}\# 7\overline{\pm}\llcorner^{f}x\vee\backslash$metric)$\sigma$$\ovalbox{\tt\small REJECT}_{\mathrm{n}}\mathrm{A}l_{arrow}^{\sim}$}$\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{i}p\backslash \backslash \backslash \vee\supset f.’.

2\downarrow\emptyset ffi\mathrm{x}\Re\theta^{\backslash }\backslash ^{\backslash }\hslash 7\pm-\mathrm{T}^{\text{る}}$caloric morphism.
)$\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}t_{arrow}^{-}\mathfrak{l}\mathrm{f}$Euclidean

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2. (Kelvin transformation) $\nu>0,$$\rho(r)=\frac{\nu^{2}}{r^{2}})\mathrm{i}\mathrm{E}\mathrm{f}\mathrm{f}\mathrm{i}_{\mathrm{L}}C^{\infty}-\ovalbox{\tt\small REJECT} 5\ovalbox{\tt\small REJECT} \text{と}T\text{る}$.

$\rho(r)l\mathrm{f}\rho(\frac{\iota/^{2}}{r})=\frac{r^{4}}{\nu^{2}}\rho(r)\not\in_{\mathrm{i}^{\backslash }}i\ovalbox{\tt\small REJECT} f_{-}^{-}’ t(ffi\mathrm{I}\rfloor\grave{\mathrm{x}}l\ovalbox{\tt\small REJECT}^{\backslash }$$k\emptyset \mathbb{H}$$f(t, x)=(t+a, \nu R\frac{x}{|x|^{2}})$$(\{\underline{\mathrm{B}}\mathrm{b}C>0, Rl\mathrm{f}^{-}\llcorner\Xi R\acute{4}\overline{\mathrm{T}}F^{1}\mathrm{J})$.

$\varphi(t, x)=C$

Ch caloric morphism.

3. (The dilation) $\nu>0,$$\{_{\llcorner}^{\mathrm{g}}C^{\infty}- 7\ovalbox{\tt\small REJECT}\Re \text{と}T\text{る}$,$\rho(r)\mathrm{t}\mathrm{f}\rho(\nu r)=\nu^{k}\rho(r)\text{を}\backslash \mathrm{f}\mathrm{f}\mathrm{i}f.’\vee t(\mathrm{f}\mathrm{f}^{1}\mathrm{J}\grave{\mathrm{x}}_{-}\}\mathrm{f}\rho(r)=r^{k})\mathrm{i}\mathrm{E}$$\mathrm{f}\emptyset \mathrm{f}\mathrm{l}$$f(t, x)=(\nu^{k+2}t+a, \nu Rx)$ ,$(\llcorner \mathrm{B}\mathrm{b}a\in \mathbb{R}, C>0, R\dagger \mathrm{f}^{-}\llcorner\Xi R\acute{(}\overline{\tau}F^{1}\mathrm{J})$I caloric morphism.
$\mathrm{f}$$k=-2\emptyset\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}\dagger \mathrm{f}t\emptyset \mathrm{f}\mathrm{f}_{\backslash }\mathrm{g}\theta^{s}\backslash 1\#=7_{J^{\vee\supset}}T\llcorner$

4. $p>0,$$\rho(r)=pr^{-2}\text{と}T\text{る}$.

,$f(t, x)=(t+a, e^{\alpha t+\beta} \frac{Rx}{|x|^{2}})$$\llcorner a,$n$\alpha,$$\beta\in \mathbb{R}(\alpha\neq 0),$$=2\emptyset\ovalbox{\tt\small REJECT}_{\mathrm{D}}^{\mathrm{A}}\mathrm{t}\mathrm{f}k\mathrm{R}^{1}\mathrm{J}\text{で},$$C>0,$\yen$\grave{\eta}$.

$\not\in^{-}\emptyset\#$$f(t, x)=(t+a, e^{\alpha t+\beta}Rx)$({El$\varphi(t, x)=C|x|^{\alpha p/2}e^{\frac{1}{4}p\alpha^{2}t}$,$R\mathfrak{l}\mathrm{f}^{-}\llcorner \mathrm{F}\mathrm{X}\acute{\mathrm{t}}\overline{\mathrm{T}}F|\mathrm{J}$)$f(t, x)=(t+a, \nu e^{\alpha t}R(bt+c)\frac{x}{|x|^{2}})$$b,$ $c,$$\alpha,$$\in \mathbb{R}(\alpha\neq 0),$$(r, \theta)\mathrm{t}\mathrm{f}\mathbb{R}^{2}\emptyset\Phi\Phi\ovalbox{\tt\small REJECT})$$\nu,$$C>0,$t$\#arrow\sim\vee\supset hT$f$\theta\dot{\backslash }_{\mathrm{Q}}\fbox\ovalbox{\tt\small REJECT}\tau \text{る}\mathrm{g}_{\mathrm{J}}\emptyset^{p\mathrm{R}}\wedge \mathrm{f}\mathrm{f}\mathrm{l}\theta^{*}\backslash$$\not\in^{-}\emptyset \mathrm{f}\mathrm{l}$$f(t, x)=(t+a, \nu e^{\alpha t}R(bt+c)x)$ ,$1^{\underline{\mathrm{B}}}\mathrm{b}a,$caloric morphism.

$\dagger \mathrm{f}$$\circ\ovalbox{\tt\small REJECT} \mathfrak{h}$$R(t)\theta^{\backslash }\backslash \acute{\pi}\mathrm{a}\mathrm{e}\backslash .C^{\backslash f_{J^{\backslash }}},$

5. $n=2,$ $p>0,$ $\rho(r)=pr^{-2}\text{と}T6$ .

,$\varphi(t, x)=C\frac{1}{|x|^{\alpha p/2}}e^{\frac{1}{4}p\alpha^{2}t}$

#G71.

(

$\varphi(t, x)=C$$\varphi(t, r, \theta)=Cr^{\alpha p/2}e^{\frac{1}{2}b\theta}e^{1_{{}_{4}\mathrm{P}^{\alpha^{2}t}}}$,$\varphi(t,r, \theta)=C\frac{1}{r^{\alpha p/2}}e^{\frac{1}{2}b\theta}e^{\frac{1}{4}p\alpha^{2}t}$$R(s)=(\begin{array}{ll}\mathrm{c}\mathrm{o}\mathrm{s}s -\mathrm{s}\mathrm{i}\mathrm{n}s\mathrm{s}\mathrm{i}\mathrm{n}s \mathrm{c}\mathrm{o}\mathrm{s}s\end{array})$Ih caloric morphism.

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[2] T. Ishihara, A mapping of Riemannian manifolds which preserves hamonic functions, J. Math.
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$\mathrm{L}\mathrm{u}\mathrm{k}\mathrm{e}\check{\mathrm{s}}$$\mathrm{e}\mathrm{d}\mathrm{s}.)$$\mathrm{J},\mathrm{V}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{l}\acute{\mathrm{y}}$