One-dimensional null-space conjecture for the Laplace–Beltrami integral equation

Let $\Gamma$ be a smooth surface, and let $\mathcal I+\mathcal R_{\text{LB}}$ denote the integral operator in~, where $\mathcal R_{\text{LB}}$ is the composition of the Laplace–Beltrami operator with $\mathcal K_{\text{LB}}$. One-dimensional null-space conjecture. The integral equation~ has only a one-dimensional null space on all smooth surfaces. The one-dimensional update described above removes this null space when the corresponding null vector has non-zero mean; the conjecture asserts that no additional null directions occur for smooth surfaces. The statement is presented without a resolution in the source.