周杰，首都师范大学

In this short seminar, we will discuss the proof of the Allard regularity theorem, which is an $\epsilon$-regularity theorem for the mean curvature equation in the Geometric Measure Theory(GMT) setting. Roughly speaking, it says if the generalized mean curvature of a rectifiable $n$-varifold belongs to $L^p$ for some supercritical index $p>n$ and the density is not less than one for any point on the support set, then at a point on the support set with mass ratio close to one, there is a neighborhood on which the support of the varifold can be written as a $C^{1,\alpha}$ graph for $\alpha=1-\frac{n}{p}$. In the first presentation, we will discuss some intuitions and then focus on the Lipschitz approximation theorem.