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

Abstract

In this paper, we prove Allard's Interior $\varepsilon$-Regularity Theorem for $m$-dimensional varifolds with generalized mean curvature in $L^p_{loc}$, $p > m$, in non-collapsed Alexandrov spaces with curvature bounded both from above and below. We first develop an intrinsic proof of the theorem for varifolds in Riemannian manifolds with metric tensor of class $\mathcal{C}^2$, without appealing to Nash's Isometric Embedding Theorem. This yields explicitly computable constants depending only on $m$, $n$, the double sided sectional curvature bounds, and the harmonic radius (or, equivalently, the injectivity radius). We then extend the result to Alexandrov spaces via the Approximation Theorem of Berestovskij and Nikolaev, where the explicit control of the constants in terms of the geometric data is required for the approximation argument.