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- W3043292271 abstract "If one considers an integral varifold $I^msubseteq M$ with bounded mean curvature, and if $S^k(I)equiv{xin M:$ no tangent cone at $x$ is $k+1$-symmetric} is the standard stratification of the singular set, then it is well known that $mathrm {dim} S^kleq k$. In complete generality nothing else is known about the singular sets $S^k(I)$. In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum $S^k(I)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $xin S^k$ that there exists a unique $k$-plane $V^k$ such that every tangent cone at $x$ is of the form $Vtimes C$ for some cone $C$. In the case of minimizing hypersurfaces $I^{n-1}subseteq M^n$ we can go further. Indeed, we can show that the singular set $S(I)$, which is known to satisfy $mathrm {dim} S(I)leq n-8$, is in fact $n-8$ rectifiable with uniformly finite $n-8$ measure. An effective version of this allows us to prove that the second fundamental form $A$ has a priori estimates in $L^7_{mathrm {weak}}$ on $I$, an estimate which is sharp as $|A|$ is not in $L^7$ for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale $r_I$ has $L^7_{weak}$-estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications $S^k_{epsilon,r}$ and $S^k_{epsilon}equiv S^k_{epsilon,0}$. Roughly, $xin S^k_{epsilon}subseteq I$ if no ball $B_r(x)$ is $epsilon$-close to being $k+1$-symmetric. We show that $S^k_epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate $mathrm {Vol}(B_r,S_epsilon^k)leq C_epsilon r^{n-k}$. The proof requires a new $L^2$-subspace approximation theorem for integral varifolds with bounded mean curvature, and a $W^{1,p}$-Reifenberg type theorem proved by the authors in [NVa]." @default.
- W3043292271 created "2020-07-23" @default.
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- W3043292271 date "2020-07-19" @default.
- W3043292271 modified "2023-10-06" @default.
- W3043292271 title "The singular structure and regularity of stationary varifolds" @default.
- W3043292271 doi "https://doi.org/10.4171/jems/987" @default.
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