Mean Curvature Blowup in Mean Curvature Flow

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Prescribing Curvatures on Three Dimensional Riemannian Manifolds with Boundaries

Let (M, g) be a complete three dimensional Riemannian manifold with boundary ∂M . Given smooth functions K(x) > 0 and c(x) defined on M and ∂M , respectively, it is natural to ask whether there exist metrics conformal to g so that under these new metrics, K is the scalar curvature and c is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a ...

Mean Curvature Flow of Higher Codimension in Hyperbolic Spaces

where H(x, t) is the mean curvature vector of Ft(M) and Ft(x) = F (x, t). We call F : M × [0, T ) → F(c) the mean curvature flow with initial value F . The mean curvature flow was proposed by Mullins [17] to describe the formation of grain boundaries in annealing metals. In [3], Brakke introduced the motion of a submanifold by its mean curvature in arbitrary codimension and constructed a genera...

Energy Estimates for a Class of Semilinear Elliptic Equations on Half Euclidean Balls

For a class of semi-linear elliptic equations with critical Sobolev exponents and boundary conditions, we prove point-wise estimates for blowup solutions and energy estimates. A special case of this class of equations is a locally defined prescribing scalar curvature and mean curvature type equation.

Blowup of Jang’s equation at outermost marginally trapped surfaces

The aim of this paper is to accurately describe the blowup of Jang’s equation. First, we discuss how to construct solutions that blow up at an outermost MOTS. Second, we exclude the possibility that there are extra blowup surfaces in data sets with non-positive mean curvature. Then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable MOTS and show exponentia...