Sharp geometric rigidity of isometries on Heisenberg groups

Abstract

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon )$-quasi-isometry on a John domain of the Heisenberg group $\mathbb H ^n, n>1,$ is close to some isometry up to proximity order $\sqrt{\varepsilon }+\varepsilon$ in the uniform norm, and up to proximity order $\varepsilon$ in the $L_p^1$-norm. We give examples showing the asymptotic sharpness of our results.

Appendix

1.1 Application of the embedding theorem

Lemma 11.

Let $f\in I(1+\varepsilon ,B(a,r))$ and

$$\begin{aligned} \Vert D_h f-I\Vert _{p,B(a,r)} \le \varepsilon |B(a,r)|^{1/p}. \end{aligned}$$

If $p>\nu$ and $f(a)=a$ then

$$\begin{aligned} \rho (f(x),x)\le C r (\sqrt{\varepsilon }+\varepsilon ) \quad \text{ for} \text{ all}\ x\in B(a,s r) \end{aligned}$$

with $s\in (0,1).$ The constant $C$ depends only on $n, p,$ and $s.$

Proof

Put $B=B(a,r).$ Denote the first $2n$ coordinates of $x^{-1}\cdot f(x)$ by $\psi (x)$ and the last coordinate of $x^{-1}\cdot f(x)$ by $\chi (x).$ Estimate $\nabla _{\mathcal L }\psi _i(x)$ for all $i=1,\dots ,2n$ and $\nabla _{\mathcal L }\chi (x).$ Clearly,

$$\begin{aligned} \Vert \nabla _\mathcal L \psi _i\Vert _{p,B}= \Vert \nabla _\mathcal L f _i-\nabla _\mathcal L x_i\Vert _{p,B} \le \Vert D_h f-I\Vert _{p,B}, \quad i=1,\dots ,2n. \end{aligned}$$

The embedding theorem (see [7] for example) yields

$$\begin{aligned} |\psi _k(x)| \le C_1 r^{1-\nu /p}\Vert \nabla _{\mathcal L }\psi _k\Vert _{p,B} \le C_2\varepsilon r\quad \text{ for} \text{ all}\ x\in \frac{s+1}{2}B, \ k=1,\dots ,2n. \end{aligned}$$

We have

$$\begin{aligned} \chi (x)= f _{2n+1}(x)-x_{2n+1} +2\sum _{j=1}^n \bigl (x_j f _{j+n}(x)- x_{j+n} f _j(x)\bigr ). \end{aligned}$$

The contact condition $X_i f(x)\in H_{f(x)}\mathbb H ^n$ for $i=1,\dots ,2n$ yields

$$\begin{aligned} X_i f_{2n+1}(x)=2\sum _{j=1}^n f_{j+n}(x)X_i f_j(x)-f_j(x)X_i f_{j+n}(x), \end{aligned}$$

and then we deduce that $\nabla _\mathcal L \chi (x)= 2\bigl ((D_h f (x))^t+I\bigr ) J \psi (x),$ where $J$ is the $2n\times 2n$ matrix defined in (8).

Applying the embedding theorem once again, we obtain

$$\begin{aligned} |\chi (x)| \le C_3 r^{1- \nu /p}\Vert \nabla _{6\mathcal L }\chi \Vert _{p,\frac{s+1}{2}B} \le C_4 r \Vert \psi \Vert _{C(\frac{s+1}{2}B)} (2+\varepsilon ) \le C_5 r^{2} \varepsilon (2+\varepsilon ) \end{aligned}$$

for all $x\in s B.$ Hence, $\rho (f(x),x)\le C_6 r(\sqrt{\varepsilon (2+\varepsilon )}+\varepsilon ) \le C_7 r(\sqrt{\varepsilon }+\varepsilon )$ for all $x\in B(a,s r).$

 

1.2 Isometries on the balls

 

Lemma 12.

If $\varphi$ is an isometry on $\mathbb H ^n$ with $\rho (\varphi (x),x)\le \varepsilon r$ for all $x\in \overline{B(a,r)}\subset \mathbb H ^n$ with $\varepsilon <1/2,$ then $\rho (\varphi (x),x)\le 5 \varepsilon s r$ for all $x\in \overline{B(a,sr)}, s\ge 1.$

 

Proof

Assume that $B(a,r)=B(0,1).$ Suppose firstly that $\varphi =\iota \circ \pi _\mathbf a \circ \varphi _A$ where $\mathbf a =(a,\alpha )\in \mathbb H ^n$ with $a\in \mathbb C ^n$ and $\alpha \in \mathbb R ,$ as well as $A\in U(n).$ If $x=0$ then $|a|<1/2$ and $|\alpha |\le 1/4.$ If $z=0$ and $t=1$ then we arrive at a contradiction:

$$\begin{aligned} 1/2 \ge \rho \bigl (\varphi (0,1),(0,1)\bigr ) = \rho \bigl ((\overline{a},-\alpha -2)\bigr ) \ge \sqrt{2+\alpha }. \end{aligned}$$

Thus, $\varphi =\pi _\mathbf a \circ \varphi _A,$ where $\mathbf a =(a,\alpha )\in \mathbb H ^n, a\in \mathbb C ^n, \alpha \in \mathbb R ,$ and $A\in U(n).$ We have

$$\begin{aligned} x^{-1}\cdot \mathbf a \cdot \varphi _A x= (-z+a+Az, \alpha +2\text{ Im}\langle a,Az\rangle -2 \text{ Im} \langle z, a+Az\rangle ). \end{aligned}$$

Clearly, $|\mathbf a |=\rho (\varphi (0),0)\le \varepsilon$ and $|Az-z|\le |Az-z+a|+|a|\le 2\varepsilon .$

We have

$$\begin{aligned} 2|\text{ Im}\langle 2a+ A z,z\rangle |&\le |\alpha +2\text{ Im}\langle a, A z\rangle - 2\text{ Im}\langle z, Az+a\rangle |\\&+ |\alpha |+ 2|\text{ Im}\langle a, A z-z+a\rangle | \le 4\varepsilon ^2 \quad \text{ for} \text{ all}\ z\in \mathbb C ^n,\ |z|\le 1. \end{aligned}$$

Suppose that $Aa=\xi a~+d$ and $(d,a)=0$ with $\xi \in \mathbb C$ and $d\in \mathbb C ^n.$ Put $z= \gamma a, |z|\le 1.$ Then

$$\begin{aligned} |\text{ Im}\langle 2a+ A z,z\rangle |= |\text{ Im}\langle 2a+\gamma (\xi a+d),\gamma a\rangle | = |a|^2|\text{ Im}(2\overline{\gamma }+ \xi |\gamma |^2)| \le 2\varepsilon ^2. \end{aligned}$$

Suppose that $a\ne 0.$ For $\gamma =\frac{1}{|a|},$ we infer that $|\text{ Im}\langle 2a+ A z,z\rangle |=|\text{ Im} \xi |\le 2\varepsilon ^2.$ For $\gamma =\frac{-i}{|a|},$

$$\begin{aligned} 2|a|\le |a|^2|\text{ Im}(2\overline{\gamma }+ \xi |\gamma |^2)| +|\text{ Im} \xi | \le 4\varepsilon ^2. \end{aligned}$$

Hence,

$$\begin{aligned} |\text{ Im}\langle A z,z\rangle | \le |\text{ Im}\langle 2a+ A z,z\rangle |+|\text{ Im}\langle 2a,z\rangle | \le 6\varepsilon ^2. \end{aligned}$$

In the case of $a=0,$ we obviously have $|\text{ Im}\langle A z,z\rangle | \le 2\varepsilon ^2.$

Consider $y=\delta _s x\in B(0,s).$ We obtain

$$\begin{aligned} y^{-1}\cdot \mathbf a \cdot \varphi _A y =(-sz+a+sAz, \alpha +2\text{ Im}\langle a,A s z\rangle -2 \text{ Im} \langle sz, a+Asz\rangle ). \end{aligned}$$

Then $|-sz+a+sAz|\le s|Az-z|+a\le (2s+1)\varepsilon$ and

$$\begin{aligned}&|\alpha +2\text{ Im}\langle a,A s z\rangle -2 \text{ Im} \langle sz, a+Asz\rangle | \\&\qquad \le |\alpha |+2|\text{ Im}\langle a,Asz+sz\rangle | +2|\text{ Im} \langle sz,Asz\rangle | \le (1+8s+12s^2)\varepsilon ^2. \end{aligned}$$

Thus, $\rho (\pi _\mathbf a \circ \varphi _A(y),y) \le 5s\varepsilon .$

Now, take an arbitrary ball $B(a,r)$ and suppose that $\rho (\varphi (x),x)\le \varepsilon r$ on $B(a,r).$ The isometry $\theta =\delta _{1/r}\circ \pi _{-a}\circ \varphi \circ \pi _{a}\circ \delta _{r}$ satisfies $\rho (\theta (y),y)\le 5s\varepsilon$ for all $y\in B(0,s).$ Inserting $x=a\cdot \delta _r y$ for $x\in B(a,sr),$ we obtain the required estimate.

The following lemma is obvious.

Lemma 13.

If $\rho (bx,x)\le \varepsilon$ on $B(a,r)$ then $\rho (bx,x)<3s\varepsilon$ on $B(a,sr), s\ge 1.$