Convergence of inflated dynamic Laplacian minimisers to material solutions

Let $M$ be the spatial manifold, let $[0,\tau]\times M$ be the spacetime manifold $\mathbb{M}_0$, and for each $a\geq 0$ let $S_a$ and $H_a$ denote the Sobolev and Cheeger constants on $\mathbb{M}_0$ with respect to the metric $G_{0,a}$. Let $s^D$ and $h^D$ be the corresponding dynamic Sobolev and Cheeger constants on $M$. For each $a$, let $F_a:\mathbb{M}_0\to\mathbb{R}$ minimise $S_a$ and let \reflectbox{\rotatebox[origin=c]{180}{\mathbb L}}_a minimise $H_a$; let $f:M\to\mathbb{R}$ minimise $s^D$ and let $\Gamma$ minimise $h^D$. Convergence conjecture.

$$\lim_{a\to\infty} S_a=s^D \quad\text{and}\quad \lim_{a\to\infty} H_a=h^D.$$

Moreover,

$$\lim_{a\to\infty}F_a(t,x)=f(x)$$

for all $t\in[0,\tau]$ and $x\in M$, and

\lim_{a\to\infty}\reflectbox{\rotatebox[origin=c]{180}{$\mathbb L$}}_a=[0,\tau]\times\Gamma.

The conjecture asserts that, as the parameter linking the time fibres tends to infinity, the spacetime minimisers become completely material and recover the corresponding spatial minimisers. The preceding proposition establishes that $H_a=S_a$ and that these quantities are nondecreasing and bounded above by the spatial constants, but the stated limiting convergence is not established in the supplied text.