Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics $\phi_1 g$ and $\phi_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres and balls, the metric $g$ is not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds $(M,g)$ we also show that the functions $\phi_1$ and $\phi_2$ are isospectral potentials for the Schr\"odinger operator $\hbar^2\Delta +\phi$. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.Comment: 34 pages, AMS-TeX; revised subsection 5.

Isospectral deformations of closed Riemannian manifolds with different scalar curvature

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.Comment: amstex, 10 pages, no figure

THE PREPARATION AND PROPERTIES OF MACROPHAGE-L CELL HYBRIDS

The plasma membrane of the mouse peritoneal macrophage has specific receptors which enable the cell to bind IgG or complement-coated sheep red cells and is also rich in a divalent cation-dependent adenosine triphosphatase (ATPase) activity. L cells lack these macrophage membrane markers. The question of macrophage membrane receptor expression was investigated in DBA/2 mouse macrophage x mouse LMTK- cell hybrids produced with the aid of Sendai virus. Three independent clones and one mass culture were isolated by their ability to grow in hypoxanthine, aminopterin, and thymidine (HAT) selection medium. These hybrids retained 85–100% of the sum of two parent cells' chromosomes and expressed several genes derived from both parents, including glucose phosphate isomerase isozymes and H-2 antigens. The hybrids displayed ATPase activity which was intermediate between that of the macrophage and L cell. The macrophage specific receptors for antibody or complement-coated red cells could not be demonstrated on hybrid cells. The selective absence of these receptors is probably because of a failure in gene expression rather than to loss of genes