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The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. (English) Zbl 1431.49046

If \(\{M_i\}\) is a sequence of \(m\)-dimensional minimal varieties in a subset \(\Omega\subset\mathbb{R}^{m+1}\) with mean curvature bounded by some \(h<\infty\) such that the measures of the boundaries of \(M_i\) are uniformly bounded on compact subsets, that is, \[ \limsup_{i\to\infty}\mathcal{H}^{m-1}(\partial M_i\cap K)<\infty \tag{*} \] for every compact subset \(K\) of \(\Omega\), then the set \(Z\) of points at which the areas of the \(M_i\) blow up is defined as \[ Z=\{x\in\Omega: \limsup_i\mathcal{H}^{m}(M_i\cap{B}_r(x))=\infty\} \] for every \(r>0\). \(Z\) is the smallest closed subset of \(\Omega \) such that the areas of the \(M_i\) are uniformly bounded as \(i\rightarrow \infty \) on compact subsets of \(\Omega \setminus Z\). If \(M\) is a smooth \(m\)-dimensional manifold \(M\subset\mathbb{R}^{m+1}\) with a normal \(\nu_M\), then for every open set \(\Omega\subset\mathbb{R}^{m+1}\) an anisotropic energy is defined as \(\mathbf{F}(M,\Omega)=\int_{M\cap\Omega}F(x,\nu_M) d\mathcal{H}^m\). A manifold \(M\) is said to be \(F\)-stationary if \(\frac{d}{dt}\mathbf{F}(\varphi_t(M),\Omega)_{|t=0}=0\), and \(M\) is called \(F\)-stable if \(\frac{d^2}{dt^2}\mathbf{F}(\varphi_t(M),\Omega)_{|t=0}\ge 0\), for every one-parameter family \(\varphi_t(x)=x+t g(x)\) of diffeomorphisms generated by a vector field \(g\in C^1_c(\Omega,\mathbb{R}^{m+1})\). If \(Z\) is a closed subset of \(\Omega\), \(f:\Omega\to\mathbb{R}\) is a \(C^2\) function such that \(f_{|Z}\) has a local maximum at \(p\), and \(\mathrm{Trace}_m(D^2f(p))\le h|Df(p)|\), where \(\mathrm{Trace}_m(D^2f(p))\) is the sum of the \(m\) lowest eigenvalues of the Hessian of \(f\) at \(p\), then \(Z\) is called a \((m,h)\)-set. In [J. Differ. Geom. 102, No. 3, 501–535 (2016; Zbl 1341.53094)], B. White found natural conditions implying that the area blow-up set \(Z\) is empty. If \(Z\) is empty, then the areas of the \(M_i\) are uniformly bounded on all compact subsets of \(\Omega\). Also, the author showed that \(Z\) belongs to the class of \((m,h)\)-sets.
In this paper, the authors extend these results to co-dimension one manifolds. They show that if \(\{M_i\}_i\) is a sequence of \(F\)-stable \(m\)-dimensional manifolds satisfying \((^\ast)\), then the area-blow up set \(Z\) is an \((m,h)\)-set in \(\Omega\) with respect to \(F\). Moreover, they prove that if \(\Omega\subset\mathbb{R}^3\) is uniformly convex, \(F\) is a uniformly elliptic integrand, \(\Gamma\subset\Omega\) is a \(C^{2,\alpha}\) embedded curve, \(M\) is an \(F\)-stable \(C^2\) 2-dimensional embedded surface in \(\Omega\) such that \(\partial M=\Gamma\), then there exist a constant \(C > 0\) and a radius \(r_1 > 0\) depending only on \(F\), \(\Omega\), \(\Gamma\) such that \(\sup\limits_{\substack{p\in\Omega \\ \mathrm{dist}(p,\Gamma) < r_1}}r_1|A_M(p)| \le C\), where \(A_M\) is the second fundamental form of \(M\). Furthermore, the constants are uniform as long as \(\Gamma\), \(\Omega\), and \(F\) vary in compact subsets of, respectively, embedded \(C^{2,\alpha}\) curves, uniform convex domains, and uniformly convex \(C^2\) integrands.

MSC:

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35D40 Viscosity solutions to PDEs

Citations:

Zbl 1341.53094
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