Discreteness conjecture for immersed submanifolds with bounded mean curvature

Let $\varphi:M\rightarrow\mathbb{R}^n$ be a bounded immersed submanifold of dimension $k\geq2$, contained in a relatively compact domain $\Omega$ with diameter $R$. Let $\mathbf{H}$ be the mean curvature vector of $M$, and suppose that

$$R\|\mathbf{H}\|_\infty<k.$$

Assume either that $\mathscr{H}^k(\lim\varphi)=0$, or that $\mathscr{H}^k(\lim\varphi\cap\Omega)=0$ and the second fundamental form $\mathrm{II}_{\partial\Omega}$ of $\partial\Omega$, oriented in the inward direction, satisfies

$$\inf_{\partial\Omega}\mathcal{P}_k^-(\mathrm{II}_{\partial\Omega})>\|\mathbf{H}\|_\infty,\qquad \inf_{\partial\Omega}\mathcal{P}_{k-1}^-(\mathrm{II}_{\partial\Omega})>-\infty$$

in the barrier sense.

Discreteness conjecture. Under either of these alternatives, the spectrum of the Laplace–Beltrami operator on $M$ is discrete.

This is presented as the geometric counterpart of the paper’s principal-eigenvalue conjecture. It concerns when small $k$-dimensional Hausdorff measure of the limit set, together with the stated curvature condition in the second alternative, forces spectral discreteness.