We consider mean curvature flow of an initial surface that is the graph a function over some domain definition in $R^n$. If not complete then we impose constant Dirichlet boundary condition at surface. establish longtime-existence and investigate projection flowing onto $R^n$, shadow flow. This moving can be seen as weak solution for hypersurfaces $R^n$ with condition. Furthermore, provide lemm...

We introduce Lagrangian mean curvature flow with boundary in Calabi–Yau manifolds by defining a natural mixed Dirichlet-Neumann condition, and prove that under this flow, the condition is preserved. also study detail of equivariant discs on Lawlor neck self-shrinking Clifford torus, demonstrate long-time existence convergence first instance rescaled second.

We prove that for the mean curvature flow of closed embedded hypersurfaces, intrinsic diameter stays uniformly bounded as approaches first singular time, provided all singularities are neck or conical type. In particular, assuming Ilmanen’s multiplicity one conjecture and no cylinder conjecture, we conclude in two-dimensional case always bounded. also obtain sharp $$L^{n-1}$$ bound curvature. T...