In the simplified form used in this course, let $V$ be an integral $m$-varifold in $B(x_0,r) \subset \mathbb R^n$ with generalized mean curvature $H \in L^p(\|V\|)$ for some $p > m$, and suppose $x_0 \in \operatorname{spt} V$. For an $m$-dimensional linear subspace $P \subset \mathbb R^n$, define the affine base ball $B_P(x_0,\rho):=(x_0+P)\cap B(x_0,\rho)$ and the corresponding cylinder $B_P(x_0,\rho)+P^\perp:=\{y+z:y\in B_P(x_0,\rho), z\in P^\perp\}$. There exist constants $\varepsilon > 0$, $\theta \in (0,1/2)$, and $\alpha \in (0,1)$, depending only on $m,n,p$, such that the following holds. Assume that $\Theta^m(\|V\|,x_0)=1$ and that there is an $m$-dimensional linear subspace $P \subset \mathbb R^n$ for which

\begin{align*} r^{-m}\|V\|(B(x_0,r)) \le (1+\varepsilon)\omega_m \end{align*}

and

\begin{align*} r^{-m}\int_{B(x_0,r)} |\pi_{T_xV}-\pi_P|^2\,d\|V\|(x) < \varepsilon \end{align*}

hold, together with

\begin{align*} r^{1-m/p}\|H\|_{L^p(\|V\|,B(x_0,r))} < \varepsilon, \end{align*}

and the height excess over the affine plane $x_0+P$ satisfies

\begin{align*} r^{-m-2}\int_{B(x_0,r)} \operatorname{dist}(x-x_0,P)^2\,d\|V\|(x) < \varepsilon. \end{align*}

Then, after rotating coordinates so that $P=\mathbb R^m \times \{0\}$, there is a function $u \in C^{1,\alpha}(B_P(x_0,\theta r);\mathbb R^{n-m})$ such that

\begin{align*} \operatorname{spt} V \cap \bigl(B_P(x_0,\theta r)+P^\perp\bigr) = \{y+u(y) : y \in B_P(x_0,\theta r)\}. \end{align*}

In this cylinder the varifold has multiplicity one on the graph.