Category Archives: Hyperbolic Geo

Hypersurface in Hyperbolic space and its tangent horospheres

Hypersurface in hyperbolic space and its tangent horospheres

Hypersurface in hyperbolic space and conformal metric on Sphere

Suppose $\Omega$ is a domain in $\mathbb{S}^n$, $g=e^{2\rho}g_0$, where $g_0$ is the standard metric on $\mathbb{S}^n$. One can construct a hypersurface in hyperboloid $\mathbb{H}^{n+1}\subset \mathbb{L}^{n+2}$ by

$\phi(x)=\frac{1}{2}(1+\sigma^2+|\nabla^0\sigma|^2)\psi(x)-\sigma(0,x)-(0,\nabla^0 \sigma)$

where $\sigma=e^{-\rho}$ and $\psi=e^{\rho}(1,x)=\sigma^{-1}(1,x)$. What are the induced metric on $\phi$? One can calculate as the following,

Suppose $u\in T_x\Omega$ is a unit eigenvector associated to the eigenvalue $s$ of $Hess(\sigma)_x$. We need to calculate $d\psi_x(u)$. Firstly it is easy to see

$d\psi_x(u)=-\sigma^{-1}u(\sigma)\psi+\sigma^{-1}(0,u)$

and

$d(\nabla^0 \sigma)_x(u)=\ \nabla_u^0\nabla^0 \sigma+(d \nabla^0\sigma_{x}(u))^{\perp}=Hess(\sigma)_x(u)+\langle d \nabla^0\sigma_{x}(u),x\rangle x$
$= \ su-\langle\nabla^0\sigma_{x},u\rangle x= \ su-u(\sigma)x$

Then

$d\phi_x(u)=u(\sigma)\sigma \psi+\frac{1}{2}u(|\nabla \sigma|^2)\psi+\frac{1}{2}(1+\sigma^2+|\nabla \sigma|^2)d\psi_x(u)-\sigma(0,u)-(0,su)$

So this gives the following formula

$d\phi_x(u)=\ \left(\frac{1}{2}-\lambda\right)d\psi_x(u).$

where $\lambda=s\sigma+\frac{1}{2}(\sigma^2-|\nabla^0\sigma|^2)$. Suppose $u_i$ is the eigenvector corresponding to $s_i$ and $u_i,i=1,\cdots, n$ are orthonormal basis, then

$\ll d\phi_x(u_i),d\phi_x(u_j)\gg=\left(\frac{1}{2}-\lambda_{i}\right)\left(\frac{1}{2}-\lambda_{j}\right)\frac{1}{\sigma^{2}}\delta_{ij}$

where $\lambda_i=s_i\sigma+\frac{1}{2}(\sigma^2-|\nabla^0\sigma|^2)$. Recall that Schouten tensor has the following formula under conformal change,

$Sch_g=Sch_0+d\rho\otimes d\rho-\nabla^{2}_0\rho-\frac{1}{2}|\nabla^0\rho|^2g_0$

$=\frac{1}{2}g_0+\frac{1}{\sigma}\nabla_0^2\sigma-\frac{1}{\sigma^2}|\nabla_0\sigma|^2$

for $g=e^{2\rho}g_0$ and $Sch_0=\frac{1}{2}g_0$. Therefore the eigenvalues of $Sch_g$ are $\lambda_i$.

$\ll d\phi,d\phi\gg=\left(\frac{1}{2}g-Sch_g\right)^2{\sigma^{2}}\delta_{ij}$

$\textbf{Remark:}$ I am using the thesis of Dimas Percy Abanto Silva and some communication with him.

Hyperbolic translation in Poincare disc model

Hyperboloid model ${\mathbb{H}^{n+1}=(x_0,x_1,\cdots,x_{n+1})}$ is hyperquadric in ${\mathbb{L}^{n+2}}$, with

$\displaystyle -x_0^2+\sum_{i=1}^{n+1} x_i^2=-1$

Half-space model ${\mathbb{R}^{n+1}_+=\{(y_1,y_2,\cdots,y_{n+1}),y_{n+1}>0\}}$. Denote ${(y_1,\cdots,y_{n},y_{n+1})=(y,y_{n+1})}$,

Poincaré ball model ${\mathbb{B}^{n+1}=(u_1,\cdots,u_{n+1})\subset\mathbb{R}^{n+1}}$ There are some transformation formula between these hyperbolic models, say

$\Psi_{12}:\mathbb{H}^{n+1}\longmapsto \mathbb{R}^{n+1}_+$

$(x_0,\cdots,x_{n+1})\longmapsto \frac{1}{x_0+x_{n+1}}\left(x_1,\cdots,x_{n},1\right)$

$\Psi_{21}:\mathbb{R}^{n+1}_+\longmapsto \mathbb{H}^{n+1}$

$(y_1,\cdots,y_{n+1})\longmapsto \left(\frac{1+|y|^2+y_{n+1}^2}{2y_{n+1}},\frac{y}{y_{n+1}},\frac{1-|y|^2-y_{n+1}^2}{2y_{n+1}}\right)$

$\Psi_{23}:\mathbb{R}^{n+1}_+\longmapsto \mathbb{B}^{n+1}$

$(y_1,\cdots,y_{n+1})\longmapsto\frac{1}{|y|^2+(y_{n+1}+1)^2}(2y,|y|^2+y_{n+1}^2-1)$

$\Psi_{32}:\mathbb{B}^{n+1}\longmapsto\mathbb{R}^{n+1}_+$
$(u_1,\cdots, u_{n+1})\longmapsto \frac{1}{|u|^2+(u_{n+1}-1)^2}\left(2u,2(1-u_{n+1})\right)-(0,\cdots,0,1)$

On ${\mathbb{R}^{n+1}_+}$, there are vertical scaling transformations which are isometries of ${\mathbb{R}^n_+}$

$\displaystyle \tau(y_1,\cdots,y_{n+1})= (ty_1,\cdots, ty_{n+1}),\quad t>0$

On ${\mathbb{B}^{n+1}}$, they are called translation in ${e_{n+1}}$ direction through the map

$\displaystyle (u,u_{n+1})\mapsto\Psi_{23}\circ\tau\circ\Psi_{32}(u,u_{n+1})$

In component,

$\displaystyle u_i\mapsto \frac{4tu_i}{(1-t)^2|u|^2+[u_{n+1}(t-1)+t+1]^2}$
$\displaystyle u_{n+1}\mapsto \frac{(t^2-1)|u|^2+4t^2u_{n+1}+(t^2-1)(u_{n+1}-1)^2}{(1-t)^2|u|^2+[u_{n+1}(t-1)+t+1]^2}$

On $\mathbb{H}^{n+1}$,

$(x_0,x,x_{n+1})\longmapsto \Psi_{21}\circ \tau\circ\Psi_{12}(x_0,x,x_{n+1})$

In component,

$\displaystyle x_0=\frac{(1+t^2)x_0^2+(1-t^2)x_{n+1}^2}{2t}$

$x_i=x_i,i=1,\cdots,n$

$\displaystyle x_{n+1}=\frac{(1-t^2)x_0^2+(1+t^2)x_{n+1}^2}{2t}$