A Bernstein type theorem for ancient solutions to the mean curvature flow in arbitrary codimension

Abstract

We prove a Bernstein type theorem for ancient solutions to the mean curvature flow in higher codimension. More precisely, we show that for a complete ancient solution { M t } t ∈ ( − ∞ , 0 ) \{M_t\}_{t\in (-\infty ,0)} , if the mean curvature vector is uniformly bounded and the ω \omega -function for M t M_t satisfies ω ≥ ω 0 \omega \geq \omega _0 uniformly for some constant ω 0 > 1 2 \omega _0>\frac {1}{\sqrt {2}} , then M t M_t must be an affine subspace for each t ∈ ( − ∞ , 0 ) t\in (-\infty ,0) .