The Danielewski surfaces are the hypersurfaces in \(\mathbb C^3\) of the form \(D_p = \{ x y - p(z) = 0\}\), where \(p(z)\) is a polynomial with only simple zeros. Each of them is equipped with a unique algebraic volume form \(\omega\). One of the main reasons of interest for these surfaces is the fact that their automorphism groups are extremely large and share many of the properties of the automorphism group of \(\mathbb C^2\), despite of the fact that, in general, they have nontrivial topology. In [F. Kutzschebauch and A. Lind, Proc. Am. Math. Soc. 139, No. 11, 3915–3927 (2011; Zbl 1241.32022)] it was proven that on each surface \(D_p\), the group generated by shear and overshear transformations is dense in the path-connected component of \(\mathrm {Aut}_{\mathrm{hol}}(D_p)\). The question on whether a corresponding statement holds in case of volume preserving automorphisms group \(\mathrm {Aut}^\omega_{\mathrm{hol}}(D_p)\) has at the moment no answer.
In this paper the authors show that an infinitesimal version of this problem has a negative answer. More precisely, they prove that, if \(\mathrm {degree}(p) \geq 3\), the Lie algebra \(\mathrm {aut}^\omega_{\mathrm{shear}}(D_p)\), generated by the holomorphic shear vector fields (they are the only ones that give volume-preserving transformations within the group of shear and overshear transformations) is not dense in the Lie algebra \(\mathrm {aut}^\omega_{\mathrm{hol}}(D_p)\) of holomorphic volume-preserving vector fields. On the contrary, if \(\mathrm {degree}(p) = 1, 2\), the Lie algebra \(\mathrm {aut}^\omega_{\mathrm{shear}}(D_p)\) is proven to be dense in \(\mathrm {aut}^\omega_{\mathrm{hol}}(D_p)\). Since, if \(\mathrm {degree}(p) = 1\), the Danielewski surface \(D_p\) is biholomorphic to \(\mathbb C^2\), this recovers and extends previous results of Andersén and Lempert on \(\mathrm {aut}^\omega_{\mathrm{hol}}(\mathbb C^2)\).
The techniques of the proof are also used to prove a version of the algebraic volume density property for the surface \(S = \mathrm {SL}_2/N\), where \(N\) is the normalizer of a maximal torus.