In 1990, Claus Gerhardt showed that this situation is characteristic of the more general case of mean-convex star-shaped smooth hypersurfaces of Euclidean space.
In particular, for any such initial data, the inverse mean curvature flow exists for all positive time and consists only of mean-convex and star-shaped smooth hypersurfaces.
The geometric estimates in Gerhardt's work follow from the maximum principle; the asymptotic roundness then becomes a consequence of the Krylov-Safonov theorem.
As is typical of geometric flows, IMCF solutions in more general situations often have finite-time singularities, meaning that I often cannot be taken to be of the form (a, ∞).
Huisken and Ilmanen proved that for any complete and connected smooth Riemannian manifold (M, g) which is asymptotically flat or asymptotically conic, and for any precompact and open subset U of M whose boundary is a smooth embedded submanifold, there is a proper and locally Lipschitz function u on M which is a positive weak solution on the complement of U and which is nonpositive on U; moreover such a function is uniquely determined on the complement of U.
However, the elliptic and weak setting gives a broader context, as such boundaries can have irregularities and can jump discontinuously, which is impossible in the usual inverse mean curvature flow.
In the special case that M is three-dimensional and g has nonnegative scalar curvature, Huisken and Ilmanen showed that a certain geometric quantity known as the Hawking mass can be defined for the boundary of {x : u(x) < t}, and is monotonically non-decreasing as t increases.
In the simpler case of a smooth inverse mean curvature flow, this amounts to a local calculation and was shown in the 1970s by the physicist Robert Geroch.
As a consequence of Huisken and Ilmanen's extension of Geroch's monotonicity, they were able to use the Hawking mass to interpolate between the surface area of an "outermost" minimal surface and the ADM mass of an asymptotically flat three-dimensional Riemannian manifold of nonnegative scalar curvature.