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On a smooth compact manifold of dimensions three and four with totally non-umbilic boundary,imposing non-negativity assumptions on curvatures of the background metric, we establish that there exists a...
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...
We will survey some recent existence theory of closed constant mean curvature hypersurfaces using the min-max method. We hope to discuss some old and new open problems on this topic as well.
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...
We study a class of mean curvature equations −Mu = H +λup where M denotes the mean curvature operator and for p ≥ 1. We show that there exists an extremal parameter λ∗ such that this equat...
Singularities of generic mean curvature flow.
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...
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...
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces ...
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...
In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a...
In this paper, we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a Riemannian manifold with codimension d 1, which generalizes the extension t...
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...

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