Curvature estimates for minimal hypersurfaces in singular spaces
The motivating impetus for this paper is the celebrated theorem of E. Heinz, that the Gaussian curvature \(K\) of a minimal surface \(z = u(x,y)\) defined over a disk of radius \(r\) satisfies \(|K|< Cr^{-2}\). The theorem was extended by Spruck, by Shih and by the reviewer to surfaces of constant mean curvature, by Simon to surfaces of ``mean curvature type'', also by Caffarelli, Nirenberg and Spruck and by Ecker and Huisken to surfaces of variable mean curvature in higher dimensions, although the result in the latter two references is dependent on a bound for the gradient. In the present work, an analogous theorem is obtained for stable solutions in \(\mathbb{R}^n\) of the ``hanging roof'' equation \[ \text{div } {Du\over \sqrt{1 + |Du|^2}} = {\alpha \over u\sqrt{1+ |Du|^2}} , \quad \alpha > 0, \] when \(\alpha + n < 4 + 2 \sqrt{2/(n + \alpha)}\). Under these hypotheses, the author proves that if \(u(x) > 0\) is in \(C^2\) of \({\mathcal B}_r (x_0) \subset \mathbb{R}^n\), and if \(M = \text{graph}(u)\) is stable in \(B_r(\xi) \subset \mathbb{R}^{n+1}\), \(\xi = (x_0, u(x_0))\), then there exists \(\varepsilon_0 (n,\alpha) \in (0,1)\) such that for \(r \leq \varepsilon_0 \xi_{n+1}\) there holds \[ (H^2 + \alpha |A|^2) (\xi) \leq C_1(n,\alpha, {\xi_{n+1}\over r}) r^{-2}, \] where \(H\) denotes the mean curvature and \(|A|\) is length of second fundamental form. If \(r \geq \varepsilon_0 \xi_{n + 1}\) then \[ (H^2 + \alpha|A|^2)(\xi) \leq C_2 (n,\alpha,q) r^{-2} \left({r \over \xi_{n+1}}\right)^{{n + \alpha \over 2 +q}} \left[1 + \left({\xi_{n+1}\over r}\right)^\alpha \right]^{1\over 2+q} \] for all \(q \in \bigl[0, \sqrt{2/(n+\alpha)}\bigr)\). As a consequence, he obtains the result that if \(\alpha + n < 4 + 2 \sqrt{2/(n+\alpha)}\), then there is no entire, globally stable smooth solution of the equation. The author observes that solutions of the equation are minimal hypersurfaces in \(\mathbb{R}^{n+1}\) endowed with a singular metric. The results are false without the stability hypothesis, and they are false for negative \(\alpha\). The author shows that for minimizing solutions, a corresponding estimate holds also for parametric surfaces. The proof of the estimates includes a new Sobolev type inequality for stationary surfaces.
scientific article; zbMATH DE number 871776
scientific article