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Lecture Note

Local Isometric Embedding of Surfaces in $\mathbb R^3$

Qing Han


In 1873, Schlaefli discussed the local isometric embedding of Riemannian manifolds in Euclidean spaces. He conjectured that a sufficiently small neighborhood in any $n$-dimensional Riemannian manifold could be isometrically embedded in $\mathbb{R}^{s_n}$, for $s_n=n(n+1)/2$. The number $s_n$ is the right one, being the number of components of the metric tensor.

The conjecture by Schlaefli for smooth manifolds had remained open for an extended period of time, even for 2-dimensional manifolds, or surfaces. The following conjecture was reposed by Yau in 1980s and 1990s: any smooth surface always has a local smooth isometric embedding in $\mathbb{R}^3$.

In this note, we shall present in a systematic way the results concerning the local isometric embedding of surfaces in $\mathbb R^3$. Basically, there are two kinds of equations (or systems) for the isometric embedding: Darboux equations and differential systems equivalent to Gauss-Codazzi system. The Darboux equation is a fully nonlinear equation of the Monge-Ampère type. The Gauss-Codazzi system can be reduced to a first order quasilinear differential system for two unknown functions.

In order to establish the local isometric embedding, we need to prove the existence of local solutions to either Darboux equation or Gauss-Codazzi system. Both are nonlinear equations, fully nonlinear for the former and quasilinear for the latter. A crucial step here is to study the linearized equations and derive a priori estimates. Such linear equations are elliptic if Gauss curvature is positive, hyperbolic if Gauss curvature negative, and of the mixed type if Gauss curvature changes its sign. Moreover, the linearized equations are degenerate where Gauss curvature vanishes. In this note, we shall distinguish these cases and study metrics with Gauss curvature which is positive, negative, nonnegative, nonpositive, or of the mixed sign. Considering the nature of the linearized equations, it is necessary to treat different cases separately. It is unlikely that there exists a unified approach.

The topic of the local isometric embedding of surfaces in $\mathbb R^3$ provides a framework for the note. We believe the note may also be useful for those interested in the degenerate equations and the equations of the mixed type.