A BRIEF HISTORY
x m
quasilinear differential system equivalent to the Gauss-Codazzi system and proved
the local isometric embedding for a large class of metrics with nonpositive Gauss
curvature. They established the isometric embedding if some directional derivative
of the Gauss curvature has a simple characterization for its zero set. This gives the
results of Nakamura and Hong as special cases.
On the other hand, Pogorelov in 1972 constructed a C 2 , 1 metric g in B\ C R2
with a sign-changing Gauss curvature such that (Br,g) cannot be realized as a
C2
surface in R3 for any r 0.
3. Global Isometric Embedding of Surfaces in
R3.
In 1916, Weyl posed the following problem. Does every smooth metric on
S2
with pointwise positive Gauss curvature admit a smooth isometric embedding in
R3?
The first attempt to solve the problem was made by Weyl himself. He used
the continuity method and obtained a priori estimates up to the second derivatives.
Twenty years later, Lewy solved the problem for an analytic metric g. In 1953,
Nirenberg gave a complete solution under the very mild hypothesis that the metric
g has continuous fourth derivatives. The result was extended to the case of con-
tinuous third derivatives of the metric by Heinz in 1962. In a completely different
approach to the problem, Alexandroff in 1942 obtained a generalized solution of
Weyl's problem as a limit of polyhedra. The regularity of this generalized solution
was proved by Pogorelov in the late 1940's. In 1994 and 1995, Guan and Li, and
Hong and Zuily independently generalized Nirenberg's result for metrics on
S2
with
nonnegative Gauss curvature.
The study of negatively curved surfaces in
R3
is closely related to the interpre-
tation of non-Euclidean geometry. The investigation of the isometric immersion of
metrics with negative curvature goes back to Hilbert. He proved in 1901 that the
full hyperbolic plane cannot be isometrically immersed in
R3.
A next natural step
is to extend such a result to complete surfaces whose Gauss curvature is bounded
above by a negative constant. The final solution of this problem was obtained by
Efimov in 1963, more than sixty years later. Efimov proved that any complete
negatively curved smooth surface does not admit a C2 isometric immersion in R3 if
its Gauss curvature is bounded away from zero. Efimov's proof is very delicate and
complicated. In the years following, Efimov extended his result in several ways.
Before the 1970's, the study of negatively curved surfaces was largely directed
at nonexistence of isometric immersions in R3. As to existence, no result for com-
plete negatively curved surfaces was known. In the 1980's, Yau proposed to find
a sufficient condition for complete negatively curved surfaces to be isometrically
immersed in R3. In 1993, Hong proved that complete negatively curved surfaces
can be isometrically immersed in
R3
if the Gauss curvature decays at a certain
rate at infinity. His discussion was based on a differential system equivalent to the
Gauss-Codazzi system.
Closely related to the global isometric embedding problem is the rigidity ques-
tion. The first rigidity result was proved by Cohn-Vossen in 1927; this states that
any two closed isometric analytic convex surfaces are congruent to each other. His
proof was later considerably shortened by Zhitomirsky. In 1943, Herglotz gave a
very short proof of the rigidity, assuming that the surfaces are three times continu-
ously differentiable. Finally in 1962 it was extended by Sacksteder to surfaces with
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