Deforming two-dimensional graphs in R4 by forced mean curvature flow

Abstract

A surface Σ₀ is a graph in R⁴ if there is a unit constant 2-form w in R⁴ such that ‹e₁ ∧ e₂, w› ≥ v₀ > 0, where {e₁, e₂} is an orthonormal frame on Σ₀. In this paper, we investigate a 2-dimensional surface Σ evolving along a mean curvature flow with a forcing term in direction of the position vector. If v₀ ≥ ${1 \over \sqrt {2}}$ holds on the initial graph Σ₀ which is the immersion of the surface Σ, and the coefficient function of the forcing vector is nonnegative, then the forced mean curvature flow has a global solution, which generalizes part of the results of Chen-Li-Tian in [2].