Weyl's embedding problem

The Weyl embedding problem is a classic problem of differential geometry and was first in 1916 by Hermann Weyl formulated. It has occupied generations of mathematicians and was solved in 1953 by Louis Nirenberg using a Monge-Ampère equation , but under very restrictive conditions. Weyl's embedding problem is very similar to the Minkowski problem .

formulation

For every positively definite quadratic form with positive Gaussian curvature given on the unit sphere , is there a concrete realization through a surface (ie an embedding ) in , i.e. a surface that has this form as the first fundamental form ? ${\displaystyle \mathbb {R} ^{3}}$

Somewhat shorter and more abstract: Be a positive definite metric on the unit sphere with positive Gaussian curvature everywhere . Can the Riemannian manifold then be isometrically embedded in the ? ${\displaystyle \mathrm {d} s^{2}}$ ${\displaystyle S^{2}}$ ${\displaystyle (S^{2},\,\mathrm {d} s^{2})}$ ${\displaystyle \mathbb {R} ^{3}}$

solution

Lewy (1938) and later Nirenberg (1953) gave important answers to Weyl's embedding problem in a work generally regarded as groundbreaking. Heinz gave the last answer so far to Weyl's embedding problem in 1962, it is: “Yes, if the first fundamental form is three times differentiable.” It is not known whether embedding is also possible with weaker requirements for the regularity of the first fundamental form.

literature

• Hermann Weyl : About the definition of a closed convex surface through its line element. In: Quarterly publication of the Natural Research Society in Zurich. 61, 1916, ISSN  0042-5672 , pp. 148-178.
• Hans Lewy : On the existence of a closed convex surface realizing a given Riemannian metric. In: Proceedings of the National Academy of Sciences . 24, 2, 1938, ISSN  0027-8424 , pp. 104-106.
• Louis Nirenberg : The Weyl and Minkowski problems in differential geometry in the large. In: Communications on Pure and Applied Mathematics. 6, 3, 1953, ISSN  0010-3640 , pp. 337-394, doi : 10.1002 / cpa.3160060303 .
• Erhard Heinz : On Weyl's embedding problem. In: Journal of Mathematics and Mechanics. 11, 3, 1962, ISSN  0095-9057 , pp. 421-454.