Allard's regularity Theorem (Theorem 5.2, https://web.stanford.edu/class/math285/ts-gmt.pdf) asserts that a varifold can be locally written as a graph. Moreover, it gives a lower bound on the size of the domain in which this is possible. The size depends on the size of subsets of varifolds where balls approximate the size of Euclidean balls of the same dimension.
I am not well versed in geometric measure theory and would like this Theorem for submanifolds of Euclidean space but I can't find it formulated anywhere. The abstract in this paper (https://arxiv.org/pdf/1311.3963.pdf) says the following:
"In the special case when V is a smooth manifold translates to the following: If $\omega^{-1} \rho^{-n}Area(V \cap B_{\rho}(x))$ is sufficiently close to 1 and the unit normal of V satisfies a $C^{0,\alpha}$ estimate, then $ V \cap B_{\rho}(x)$ is the graph of a $C^{1,\alpha}$ function with estimates."
I would like a theorem of the following form.
Suppose $V$ is a closed smooth submanifold of $\mathbb{R}^n$ such that the second fundamental form of $V$ is bounded by $C$ and its derivatives by $C^2$ with positive injectivity radius $r_V$. Then there is a radius $R(C,r_V)>0$ such that for any $p \in V$ there is a function $u_p$ $$ B_R(p) \cap V = graph(u_p) $$ where $B_R(p)\subset \mathbb{R}^n$
Edit: For anyone interested, I found the answer to my question in Theorem 3.8 of https://people.math.sc.edu/howard/Notes/schur.pdf
I found the a source that asserts that a domain, wherein a submanifold $V$ of Euclidean space can be written as a graph, contains a ball whose radius only depends: Proposition 3.8 in
https://people.math.sc.edu/howard/Notes/schur.pdf
The Theorem is formulated for submanifolds whose second fundamental form is bounded by $1$ but after a rescaling argument it follows that the radius depends only linearly on the norm of the second fundamental form.