Convex affine maximal surfaces
type: Item
instance of: scholarly article
title: Convex affine maximal surfaces
zbMATH Open document ID: 0653.53006
DOI: 10.1007/BF03323241
author: Eugenio Calabi
published in: Results in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: This paper deals with affine maximal surfaces. The author derives the fundamental formulas of affine differential geometry for locally strictly convex surfaces, expressed in terms of complex isothermal parameters with respect to the Blaschke metric, and proves that an affine maximal surface can be generated by a holomorphic immersion \(Z: M\to C^ 3\) satisfying \(\det(Z-\bar Z,Z',\bar Z')>0\) under the relation \[ x=-i(Zx\bar Z+\int ZxdZ-\int \bar Zxd\bar Z). \] This representation was also obtained by \textit{C.-L. Terng} [Ann. Math. Stud. 103, 207-216 (1983; Zbl 0533.53004)]. Using the complex representation the author proves the following theorem. Let \(x: M\to A^ 3\) be locally strictly convex, maximal, and complete with respect to the Blaschke metric. If the image of the holomorphic curve Z lies in a half space of \(C^ 3\) then x(M) lies in an elliptic paraboloid. This theorem is an improvement of his previous result in Am. J. Math. 104, 91-126 (1982; Zbl 0501.53037).
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 4065829
zbMATH Keywords: affine maximal surfaces
zbMATH Keywords: locally strictly convex surfaces
zbMATH Keywords: holomorphic immersion
zbMATH Keywords: complex representation
zbMATH Keywords: Blaschke metric
MaRDI profile type: Publication
cites work: An extension of E. Hopf's maximum principle with an application to Riemannian geometry
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cites work: On the inequality \(\Delta u \geqq f (u)\)
description: scientific article; zbMATH DE number 4065829