Strictly convex curves, convex hulls and surfaces of positive Gauss curvature.

*(English)*Zbl 0939.53034
Amann, H. (ed.) et al., Progress in partial differential equations. Papers from the 3rd European conference on elliptic and parabolic problems, Pont-à-Mousson, France, June 1997. Vol. 2. Harlow: Longman. Pitman Res. Notes Math. Ser. 384, 138-142 (1998).

This is a survey paper concerning the fundamental and difficult question of determining when a smooth collection of Jordan curves \(\Gamma= (\Gamma_1,\dots, \Gamma_m)\) in \(\mathbb{R}^3\) (or, more generally, a collection of disjoint codimension 2 closed embedded submanifolds of \(\mathbb{R}^{n+1}\)) bounds an embedded hypersurface of strictly positive Gauss curvature. First, the case when \(\Gamma\) is the graph of a function (or, more generally, a radial graph) over the boundary of a domain \(\Omega\subset \mathbb{R}^n \) (resp. \(S^n\)) is considered. The prescribed Gauss curvature condition can be expressed in these situations as Monge-Ampère type boundary value problems. The possibility to allow \(\Omega\) of arbitrary geometry is discussed (see the joint papers of the author with B. Guan [Ann. Math., II. Ser. 138, 601-624 (1993; Zbl 0840.53046)] and H. Rosenberg [J. Differ. Geom. 40, 379-409 (1994; Zbl 0823.53047)]).

Secondly, the case of a single smooth closed codimension 2 embedded submanifold \(\Gamma\) of \(\mathbb{R}^{n+1}\) is considered. This corresponds to the geometric question of the existence of convex hypersurfaces \(S^{\pm}\) with Gauss curvature \(K(S^{\pm})\equiv 0\). The hypersurfaces \(S^{\pm}\) correspond to the boundaries of the convex hull \(\mathcal{C}(\Gamma)\) which is the convex region bounded by the two “convex caps” \(S^{\pm}\) which meet along \(\Gamma\). For \(\Gamma\), a graph of a function \(\varphi\) over the boundary \(\partial\Omega\) of a strictly convex domain \(\Omega\), the lower cap \(S\) can be represented as a graph \(x _{n+1}=u(x)\), where \(u(x)= \max\{v(x) \mid v\in C^{0}(\overline\Omega)\), \(v\) convex, \(v\leq\varphi\) on \(\partial \Omega\}\). The convex function \(u\) is a weak solution (in the Alexandrov sense) of a degenerate Monge-Ampère boundary value problem.

The problem of regularity of \(u\) is discussed, including the optimal regularity established by L. Caffarelli, L. Nirenberg and the author [Rev. Mat. Iberoam. 2, 19-27 (1987; Zbl 0611.35029)]. The important generalization of this last result obtained by B. Guan [Trans. Am. Math. Soc. 350, 4955-4971 (1998; Zbl 0919.35046)] as well as the extension of Guan’s result to space curves (or codimension 2 submanifolds) are pointed out.

For the entire collection see [Zbl 0905.00060].

Secondly, the case of a single smooth closed codimension 2 embedded submanifold \(\Gamma\) of \(\mathbb{R}^{n+1}\) is considered. This corresponds to the geometric question of the existence of convex hypersurfaces \(S^{\pm}\) with Gauss curvature \(K(S^{\pm})\equiv 0\). The hypersurfaces \(S^{\pm}\) correspond to the boundaries of the convex hull \(\mathcal{C}(\Gamma)\) which is the convex region bounded by the two “convex caps” \(S^{\pm}\) which meet along \(\Gamma\). For \(\Gamma\), a graph of a function \(\varphi\) over the boundary \(\partial\Omega\) of a strictly convex domain \(\Omega\), the lower cap \(S\) can be represented as a graph \(x _{n+1}=u(x)\), where \(u(x)= \max\{v(x) \mid v\in C^{0}(\overline\Omega)\), \(v\) convex, \(v\leq\varphi\) on \(\partial \Omega\}\). The convex function \(u\) is a weak solution (in the Alexandrov sense) of a degenerate Monge-Ampère boundary value problem.

The problem of regularity of \(u\) is discussed, including the optimal regularity established by L. Caffarelli, L. Nirenberg and the author [Rev. Mat. Iberoam. 2, 19-27 (1987; Zbl 0611.35029)]. The important generalization of this last result obtained by B. Guan [Trans. Am. Math. Soc. 350, 4955-4971 (1998; Zbl 0919.35046)] as well as the extension of Guan’s result to space curves (or codimension 2 submanifolds) are pointed out.

For the entire collection see [Zbl 0905.00060].

Reviewer: Mircea Craioveanu (Timişoara)

##### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |