Allard's interior $\varepsilon$-Regularity Theorem in Alexandrov spaces

In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, for $p \in \mathbb{R}$ such that $p>m$, in Alexandrov spaces of dimension $n$ with double-sided bounded intrinsic sectional curvature. We first give an intrinsic proof of this theorem in the case of varifolds in Riemannian manifolds of dimension $n$ whose metric tensor is at least of class $\mathcal{C}^2$, without using Nash's Isometric Embedding Theorem. This approach provides explicitly computable constants that depend only on $n$, $m$, the injectivity radius and bounds on the sectional curvature, which is essential for proving our main theorem, as we establish it through a density argument in the topological space of Riemannian manifolds with positive lower bounds on the injectivity radius and double-sided bounds on sectional curvature, equipped with the $\mathcal{C}^{1,\alpha}$ topology, for every $\alpha \in ]0,1[$ (in fact, it is enough with the $W^{2,q}$ topology for some suitable $q$ large enough).