Curvature estimates for minimal hypersurfaces in singular spaces
type: Item
instance of: scholarly article
title: Curvature estimates for minimal hypersurfaces in singular spaces
zbMATH Open document ID: 0852.53011
DOI: 10.1007/BF01231452
author: Ulrich Dierkes
published in: Inventiones Mathematicae
publication date: 1996-12-01T00:00:00Z
full work available at URL: https://eudml.org/doc/144328
review text: 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…
reviewed by: Robert Finn
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 871776
zbMATH Keywords: minimal hypersurfaces
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: mean curvature
zbMATH Keywords: second fundamental form
zbMATH Keywords: stability
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 871776