An existence theorem for Brakke flow with fixed boundary conditions

Consider an arbitrary closed, countably n-rectifiable set in a strictly convex $$(n+1)$$ ( n + 1 ) -dimensional domain, and suppose that the set has finite n-dimensional Hausdorff measure and the complement is not connected. Starting from this given set, we show that there exists a non-trivial Brakke flow with fixed boundary data for all times. As $$t \uparrow \infty $$ t ↑ ∞ , the flow sequentially converges to non-trivial solutions of Plateau’s problem in the setting of stationary varifolds.

Consequences of Strong Stability of Minimal Submanifolds

In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang [12] directly extends to the enhanced Brakke flows of Ilmanen [5]. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow, which are proved to occur by Neves [7].