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We study mean curvature flows (MCFs) coming out of cones. As cones are singular at the origin, the evolution is generally not unique. A special case of such flows is known as the self-expanders. We wi...
Mean curvature flow is the fastest way to decrease the area of surfaces. It is the model in many disciplines such as material science, fluid mechanism, and computer graphics. The translators are a spe...
Singularities of generic mean curvature flow.
We prove that any complete surface with constant mean curvature in a homogeneous space E(\kappa,\tau) which is transversal to the vertical Killing vector field is, in fact, a vertical graph. As a cons...
In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differenti...
We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching conditio...
Abstract: In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has iso...
Abstract: Consider the mean curvature flow of an (n+1)-dimensional, compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the m-th homotopy g...
Abstract: We show, for mean curvature flows in Euclidean space, that if one of the tangent flows at a given spacetime point consists of a closed, multiplicity-one, smoothly embedded self-similar shrin...
Abstract: We prove some Caccioppoli's inequalities for the traceless part of the second fundamental form of a complete, non compact, finite index, constant mean curvature hypersurface of a Riemannian ...
Abstract: Let $(M,\bar{g})$ be a K\"ahler surface with a constant holomorphic sectional curvature $k>0$, and $\Sigma$ an immersed symplectic surface in $M$. Suppose $\Sigma$ evolves along the mean cur...
It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for...

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