[proofplan]
This proof treats the stated result as the simplified corollary form of Allard's published interior [regularity theorem](/theorems/2750), not as a derivation of the full excess-improvement machinery from first principles. We first reduce the hypotheses by translation, rotation, and dilation to the unit cylinder with base plane $P_0 := \mathbb{R}^m \times \{0\}$. The normalized hypotheses are exactly the small mass ratio, tilt excess, height excess, and scaled $L^p$ mean-curvature assumptions required by the external Allard regularity input. Pulling the resulting normalized graph back through the affine normalization gives the stated graphical representation, and the multiplicity-one conclusion is preserved by the affine diffeomorphism.
[/proofplan]
[step:Normalize the ball and the reference plane]
Let $P_0 := \mathbb{R}^m \times \{0\} \subset \mathbb{R}^n$. Choose an orthogonal map $Q: \mathbb{R}^n \to \mathbb{R}^n$ satisfying $Q(P)=P_0$, and define the affine dilation $F: B(x_0,r) \to B(0,1)$ by
\begin{align*}
F(x)=r^{-1}Q(x-x_0).
\end{align*}
Let $W := F_{\#}V$ be the pushforward integral $m$-varifold in $B(0,1)$, and let $H_W \in L^p(\|W\|)$ denote its generalized mean curvature. For an integral varifold $Z$, write $T_aZ$ for the approximate tangent $m$-plane defined for $\|Z\|$-almost every point $a$, and write $\pi_{T_aZ}$ for the [orthogonal projection](/theorems/437) onto that plane. Pushforward by an affine diffeomorphism preserves integrality and sends $x_0 \in \operatorname{spt}V$ to $0 \in \operatorname{spt}W$. Since $F$ has dilation factor $r^{-1}$, the varifold weight measure transforms by
\begin{align*}
\|W\|(A)=r^{-m}\|V\|(F^{-1}(A))
\end{align*}
for every Borel set $A \subset B(0,1)$, and the mean curvature scales by
\begin{align*}
\|H_W\|_{L^p(\|W\|,B(0,1))}=r^{1-m/p}\|H\|_{L^p(\|V\|,B(x_0,r))}.
\end{align*}
Thus $W$ satisfies the normalized mass bound
\begin{align*}
\|W\|(B(0,1)) \leq (1+\varepsilon)\omega_m.
\end{align*}
It satisfies the normalized tilt bound
\begin{align*}
\int_{B(0,1)} |\pi_{T_zW}-\pi_{P_0}|^2\,d\|W\|(z) < \varepsilon.
\end{align*}
It satisfies the normalized mean-curvature bound
\begin{align*}
\|H_W\|_{L^p(\|W\|,B(0,1))} < \varepsilon.
\end{align*}
It satisfies the normalized height bound
\begin{align*}
\int_{B(0,1)} \operatorname{dist}(z,P_0)^2\,d\|W\|(z) < \varepsilon.
\end{align*}
Also $\Theta^m(\|W\|,0)=\Theta^m(\|V\|,x_0)=1$.
[guided]
The purpose of the normalization is to remove the inessential parameters $x_0$, $r$, and $P$. Define $P_0 := \mathbb{R}^m \times \{0\}$ and choose an orthogonal map $Q: \mathbb{R}^n \to \mathbb{R}^n$ with $Q(P)=P_0$. Define the affine map $F: B(x_0,r) \to B(0,1)$ by
\begin{align*}
F(x)=r^{-1}Q(x-x_0).
\end{align*}
This map translates $x_0$ to $0$, rotates $P$ onto $P_0$, and rescales length by $r^{-1}$. Let $W := F_{\#}V$ be the pushforward varifold. Because $F$ is an affine diffeomorphism, it preserves the property of being an integral varifold, and since $x_0 \in \operatorname{spt}V$, we have $0=F(x_0) \in \operatorname{spt}W$.
We now check how every scale-invariant hypothesis transforms. The weight measure satisfies
\begin{align*}
\|W\|(A)=r^{-m}\|V\|(F^{-1}(A))
\end{align*}
for every Borel set $A \subset B(0,1)$. Taking $A=B(0,1)$ gives
\begin{align*}
\|W\|(B(0,1))=r^{-m}\|V\|(B(x_0,r))\leq (1+\varepsilon)\omega_m.
\end{align*}
The tangent plane is rotated by $Q$, so the Hilbert-Schmidt distance between orthogonal projections is unchanged:
\begin{align*}
|\pi_{T_{F(x)}W}-\pi_{P_0}|=|\pi_{T_xV}-\pi_P|.
\end{align*}
Using the measure transformation, this yields
\begin{align*}
\int_{B(0,1)} |\pi_{T_zW}-\pi_{P_0}|^2\,d\|W\|(z)
=r^{-m}\int_{B(x_0,r)} |\pi_{T_xV}-\pi_P|^2\,d\|V\|(x)<\varepsilon.
\end{align*}
The same substitution gives the height excess bound because $\operatorname{dist}(F(x),P_0)=r^{-1}\operatorname{dist}(x-x_0,P)$:
\begin{align*}
\int_{B(0,1)} \operatorname{dist}(z,P_0)^2\,d\|W\|(z)
=r^{-m-2}\int_{B(x_0,r)} \operatorname{dist}(x-x_0,P)^2\,d\|V\|(x)<\varepsilon.
\end{align*}
Finally, mean curvature has units of inverse length, so the affine dilation gives
\begin{align*}
\|H_W\|_{L^p(\|W\|,B(0,1))}=r^{1-m/p}\|H\|_{L^p(\|V\|,B(x_0,r))}<\varepsilon.
\end{align*}
The density is invariant under affine dilation, hence $\Theta^m(\|W\|,0)=1$. These are precisely the normalized hypotheses needed for the local regularity theorem.
[/guided]
[/step]
[step:Apply Allard's normalized regularity theorem as the prior analytic input]
We now invoke the [Allard Regularity Theorem](/theorems/Allard%20Regularity%20Theorem) in its published interior regularity form with $L^p$ mean curvature and $p>m$; this is the independent external excess-improvement and iteration result from which the present simplified course statement is a corollary. In the notation used here, Allard's theorem gives constants $\varepsilon_A>0$, $\theta\in(0,1/2)$, $\alpha\in(0,1)$, and $C_A<\infty$, depending only on $m,n,p$, with the following consequence: if an integral $m$-varifold in $B(0,1)$ has density one at $0$, mass ratio at most $(1+\varepsilon_A)\omega_m$, tilt excess, height excess, and $L^p$ mean-curvature norm all less than $\varepsilon_A$, then in $B_{P_0}(0,\theta)+P_0^\perp$ its support is a single $C^{1,\alpha}$ graph over $P_0$, with multiplicity one. The bibliographic source is Allard's interior regularity theorem, doi:10.2307/1970595; the linked Androma theorem records the external theorem being used here. Choose the constant $\varepsilon$ in the theorem statement so that $0<\varepsilon\leq\varepsilon_A$. The normalized varifold $W$ satisfies each hypothesis verified in the preceding step, so there is a map $w: B_{P_0}(0,\theta) \to P_0^\perp$ with
\begin{align*}
w\in C^{1,\alpha}(B_{P_0}(0,\theta);P_0^\perp)
\end{align*}
such that
\begin{align*}
\operatorname{spt}W\cap\bigl(B_{P_0}(0,\theta)+P_0^\perp\bigr)
=\{y+w(y):y\in B_{P_0}(0,\theta)\},
\end{align*}
and $W$ has multiplicity one on this graph.
[guided]
This is the point at which the proof uses the deep analytic theorem. The normalization step did not prove excess decay or graphical regularity; it only checked that the hypotheses have the correct scale-invariant form. The independent input is the [Allard Regularity Theorem](/theorems/Allard%20Regularity%20Theorem), used through the following normalized consequence of its published interior regularity statement: there are constants $\varepsilon_A>0$, $\theta\in(0,1/2)$, $\alpha\in(0,1)$, and $C_A<\infty$, depending only on $m,n,p$, such that an integral $m$-varifold in $B(0,1)$ with density one at $0$, mass ratio at most $(1+\varepsilon_A)\omega_m$, tilt excess less than $\varepsilon_A$, height excess less than $\varepsilon_A$, and $L^p$ mean-curvature norm less than $\varepsilon_A$ is represented in $B_{P_0}(0,\theta)+P_0^\perp$ by one $C^{1,\alpha}$ graph over $P_0$, with multiplicity one. This is a bibliographic invocation of Allard's theorem, doi:10.2307/1970595, as represented by the linked Androma theorem; the present theorem is the scaled corollary form used in the course.
We now verify the hypotheses of that normalized consequence. The preceding step proved the normalized mass estimate
\begin{align*}
\|W\|(B(0,1)) \leq (1+\varepsilon)\omega_m.
\end{align*}
It also proved the normalized tilt estimate
\begin{align*}
\int_{B(0,1)} |\pi_{T_zW}-\pi_{P_0}|^2\,d\|W\|(z) < \varepsilon.
\end{align*}
The normalized height estimate is
\begin{align*}
\int_{B(0,1)} \operatorname{dist}(z,P_0)^2\,d\|W\|(z) < \varepsilon.
\end{align*}
The normalized mean-curvature estimate is
\begin{align*}
\|H_W\|_{L^p(\|W\|,B(0,1))} < \varepsilon.
\end{align*}
Finally, $\Theta^m(\|W\|,0)=1$. Choosing $\varepsilon$ so that $0<\varepsilon\leq\varepsilon_A$ makes each smallness hypothesis of Allard's normalized theorem valid for $W$. Therefore there exists a map $w: B_{P_0}(0,\theta) \to P_0^\perp$ satisfying
\begin{align*}
w\in C^{1,\alpha}(B_{P_0}(0,\theta);P_0^\perp)
\end{align*}
and
\begin{align*}
\operatorname{spt}W\cap\bigl(B_{P_0}(0,\theta)+P_0^\perp\bigr)
=\{y+w(y):y\in B_{P_0}(0,\theta)\}.
\end{align*}
The same normalized theorem supplies the multiplicity-one conclusion for $W$ on this graph.
[/guided]
[/step]
[step:Pull the graphical representation back to the affine base plane]
For $\rho>0$, write $B_P(x_0,\rho):=(x_0+P)\cap B(x_0,\rho)$ and $B_P(x_0,\rho)+P^\perp:=\{y+z:y\in B_P(x_0,\rho), z\in P^\perp\}$. Define the inverse-normalized graph map $u: B_P(x_0,\theta r) \to P^\perp$ by
\begin{align*}
u(y)=r Q^{-1} w(r^{-1}Q(y-x_0)).
\end{align*}
Here $y-x_0\in P$, so $r^{-1}Q(y-x_0)\in P_0$ and the formula is well-defined. Since $w\in C^{1,\alpha}(B_{P_0}(0,\theta);P_0^\perp)$ and the maps $y\mapsto r^{-1}Q(y-x_0)$ and $z\mapsto rQ^{-1}z$ are smooth linear-affine maps between the relevant affine and linear planes, the chain rule gives $u\in C^{1,\alpha}(B_P(x_0,\theta r);P^\perp)$. Applying $F^{-1}$ to the graphical identity for $W$ gives
\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*}
After the stated rotation of coordinates, $P=P_0=\mathbb{R}^m\times\{0\}$ and $P^\perp=\mathbb{R}^{n-m}$, so $u$ is viewed as an element of $C^{1,\alpha}(B_P(x_0,\theta r);\mathbb{R}^{n-m})$.
[guided]
The normalized graph must be transported back through the inverse affine map. For $\rho>0$, define
\begin{align*}
B_P(x_0,\rho):=(x_0+P)\cap B(x_0,\rho).
\end{align*}
Also define the corresponding cylinder by
\begin{align*}
B_P(x_0,\rho)+P^\perp:=\{y+z:y\in B_P(x_0,\rho),\ z\in P^\perp\}.
\end{align*}
For $y\in B_P(x_0,\theta r)$, the vector $y-x_0$ lies in $P$, hence $r^{-1}Q(y-x_0)\in P_0$ and $|r^{-1}Q(y-x_0)|<\theta$. Thus the formula
\begin{align*}
u(y)=r Q^{-1} w(r^{-1}Q(y-x_0))
\end{align*}
defines a map $u: B_P(x_0,\theta r)\to P^\perp$. Because $w\in C^{1,\alpha}(B_{P_0}(0,\theta);P_0^\perp)$ and the two maps $y\mapsto r^{-1}Q(y-x_0)$ and $z\mapsto rQ^{-1}z$ are smooth affine-linear maps between the relevant planes, the chain rule gives
\begin{align*}
u\in C^{1,\alpha}(B_P(x_0,\theta r);P^\perp).
\end{align*}
Applying $F^{-1}$ to each point $a=y_0+w(y_0)$ of the normalized graph gives
\begin{align*}
F^{-1}(a)=x_0+rQ^{-1}y_0+rQ^{-1}w(y_0).
\end{align*}
Writing $y=x_0+rQ^{-1}y_0\in B_P(x_0,\theta r)$, this becomes $F^{-1}(a)=y+u(y)$. Therefore
\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*}
After rotating coordinates so that $P=P_0=\mathbb{R}^m\times\{0\}$, we identify $P^\perp$ with $\mathbb{R}^{n-m}$ by the coordinate projection onto the last $n-m$ coordinates, so $u$ is a $\mathbb{R}^{n-m}$-valued $C^{1,\alpha}$ map.
[/guided]
[/step]
[step:Transfer multiplicity one back from the normalized graph]
Multiplicity of an integral varifold is invariant under the affine diffeomorphism $F$ because $F$ is one-to-one and maps approximate tangent planes isomorphically. Since the normalized varifold $W$ has multiplicity one on
\begin{align*}
\{y+w(y):y\in B_{P_0}(0,\theta)\},
\end{align*}
the original varifold $V=F^{-1}_{\#}W$ has multiplicity one on the pulled-back graph
\begin{align*}
\{y+u(y):y\in B_P(x_0,\theta r)\}.
\end{align*}
This is exactly the asserted graphical representation and multiplicity-one conclusion in the cylinder, completing the proof.
[/step]
