Abstract For any compact connected onedimensional submanifold $K\subset \mathbb R^{2\times 2}$ without boundary that has no rankone connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}} Du M ^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset{{\mathbb{R}}}^{2\times 2}$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold true in $\mathbb R^{n\times n}$ for $n\geq 3$.
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Boundary regularity and stability for spaces with Ricci bounded below
Abstract This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metricmeasure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N1$$ N  1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N1})$$ B 1 ( 0 N  1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N1$$ N  1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N1$$ N  1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X .
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 Award ID(s):
 1809011
 NSFPAR ID:
 10404632
 Date Published:
 Journal Name:
 Inventiones mathematicae
 Volume:
 228
 Issue:
 2
 ISSN:
 00209910
 Page Range / eLocation ID:
 777 to 891
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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