Mean Curvature Motion of Non-parametric Hypersurfaces with Contact Angle Condition

Mean Curvature Motion of Graphs with Constant Contact Angle at a Free Boundary

We consider the motion by mean curvature of an n-dimensional graph over a time-dependent domain in Rn, intersecting Rn at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved, an...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Spacelike hypersurfaces in Riemannian or Lorentzian space forms satisfying L_k(x)=Ax+b

We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

Linear Weingarten hypersurfaces in a unit sphere

In this paper, by modifying Cheng-Yau$'$s technique to complete hypersurfaces in $S^{n+1}(1)$, we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related which improve the result of [H. Li, Hypersurfaces with constant scalar curvature in space forms, {em Math. Ann.} {305} (1996), 665--672].  

Motion by volume preserving mean curvature flow near cylinders

We investigate the volume preserving mean curvature flow with Neumann boundary condition for hypersurfaces that are graphs over a cylinder. Through a center manifold analysis we find that initial hypersurfaces sufficiently close to a cylinder of large enough radius, have a flow that exists for all time and converges exponentially fast to a cylinder. In particular, we show that there exist globa...