We derive an upper bound on the waiting time for a non star-shaped hypersurface in R n+1 moving by Inverse Mean Curvature Flow to become starshaped. Combining this result with an embeddedness principle for the flow, we provide an upper bound on the maximal time of existence for initial surfaces which are not topological spheres. Finally, we establish the existence of finite-time singularities for certain topological spheres under IMCF.

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Given a non-star-shaped H > 0 rotationally symmetric hypersurface N 0 ⊂ R n+1 with a sufficiently long, thick neck, we prove that the corresponding solution N t to Inverse Mean Curvature Flow exists for all time and homothetically converges to a round sphere at large times.

For an H > 0 H>0 rotationally symmetric embedded torus N 0 ⊂ R 3 N_{0} \subset \mathbb {R}^{3} evolved by Inverse Mean Curvature Flow, we show that the total curvature | A | |A| remains bounded up to the singular time T max T_{\max } . This in turn implies convergence of the N t N_{t} to a C 1 C^{1} rotationally symmetric embedded torus N T max N_{T_{\max }} as t → T max t \rightarrow T_{\max } without rescaling, contrasting sharply with the behavior of other extrinsic flows. Later, we note a scale-invariant L 2 L^{2} energy estimate on any flow solution in R 3 \mathbb {R}^{3} that may be useful in ruling out curvature blowup near singularities more generally.

We give a simple proof to the computation of ADM mass of the static extensions of small spheres in Wiygul [13, 14]. It makes use of the mass formula $m = \frac {1}{4\pi } \int _{\partial M} \frac {\partial V}{\partial \nu }$ for an asymptotically flat static manifold with boundary.