View source for Weyl problem

The problem of realizing, in three-dimensional Euclidean space, a regular metric of positive curvature given on a sphere — i.e. the problem of the existence of a regular ovaloid for a prescribed metric. The problem was posed in 1915 by H. Weyl [[#References|[1]]]. H. Lewy

in 1937 solved the Weyl problem for the case of an analytic metric: An analytic metric of positive curvature, defined on a sphere, is always realized by some analytic surface in three-dimensional Euclidean space. A complete solution of the Weyl problem was given by the theorem of A.D. Aleksandrov [[#References|[3]]] on the realization of a metric of positive curvature by a convex surface, in conjunction with a theorem of A.V. Pogorelov on the regularity of a convex surface with a regular metric. This solution says that a regular metric of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097730/w0977301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097730/w0977302.png" />, of positive Gaussian curvature, defined on a manifold which is homeomorphic to a sphere, can be realized by a closed regular convex surface of class at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097730/w0977303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097730/w0977304.png" />. If the metric is analytic, the surface is analytic as well. Pogorelov ([[#References|[3]]], Chapt. 6) posed and solved Weyl's problem for the general case of a three-dimensional Riemann surface.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Weyl,   "Ueber die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement"  ''Vierteljahrschrift Naturforsch. Gesell. Zurich'' , '''3''' :  2  (1916)  pp. 40–72</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  H. Lewy,   "A priori limitations for solutions of Monge–Ampère equations"  ''Trans. Amer. Math. Soc.'' , '''37'''  (1935)  pp. 417–434</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  H. Lewy,   "On the non-vanishing of the Jacobian in certain one-to-one mappings"  ''Bull. Amer. Math. Soc.'' , '''42'''  (1936)  pp. 689–692</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Pogorelov,   "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1972)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Schneider,   "Boundary structure and curvature of convex bodies"  J. Tölke (ed.)  J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser  (1979)  pp. 13–59</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Nirenberg,   "The Weyl and Minkowski problems in differential geometry in the large"  ''Comm. Pure Appl. Math.'' , '''6'''  (1953)  pp. 337–394</TD></TR></table>