The harmonic mean curvature flow of nonconvex surfaces in $${\mathbb{R}^3}$$

Timelike Surfaces with Harmonic Inverse Mean Curvature

In classical differential geometry, surfaces of constant mean curvature (CMC surfaces) have been studied extensively [1]. As a generalization of CMC surfaces, Bobenko [2] introduced the notion of surface with harmonic inverse mean curvature (HIMC surface). He showed that HIMC surfaces admit Lax representation with variable spectral parameter. In [5], Bobenko, Eitner and Kitaev showed that the G...

Surfaces with Harmonic Inverse Mean Curvature and Painlev Equations

In this paper we study surfaces immersed in R such that the mean curvature function H satisfies the equation (1=H) = 0, where is the Laplace operator of the induced metric. We call them HIMC surfaces. All HIMC surfaces of revolution are classified in terms of the third Painlevé transcendent. In the general class of HIMC surfaces we distinguish a subclass of -isothermic surfaces, which is a gene...

Mean Curvature Blowup in Mean Curvature Flow

In this note we establish that finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M t →֒ (N, h) are characterised by the blow up of the mean curvature.

Mean Curvature Flow of Surfaces in Einstein Four-Manifolds

Let Σ be a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold (M,ω). We consider the evolution of Σ in the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms ω and ω that determine different orientations and Σ is symplectic with...

Flow by mean curvature of surfaces of any codimension