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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].
##### 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