While I finish my next post on the Dirichlet problem for the Monge-Ampere equation, I thought I could mention two neat things I have learned from reading Jerry Kazdan’s survey on Prescribing the curvature of a Riemannian manifold, a short book that I recommend strongly (its a bit out of date, but hey!, geometry has advanced quite a bit in the last couple of years!)

First, a cute solution to an easily-stated problem, as observed by Wallach and Warner in 1970

*Theorem: Given a compact smooth 2-d manifold and is a 2-form such that , then there exists a smooth Riemannian metric on such that , where is the Gauss curvature of and is the area form of .*

*Proof: *Pick your favorite metric on , we will prove that the metric we are looking for is in fact *conformal* to , for any smooth function , define . Using the well known formulas for the Gauss curvature and area form of , one arrives at the identity

where the sub-indexes and refer to the object corresponding to the metric and , in particular, is the Laplace operator for . Then one wants to pick such that

or

but, using the metric we can write for some (or equivalently, using the Hodge star operator given by ) thus the we want is the solution of the equation

but since we have that thus by standard elliptic theory (or Fredholm theory, etc) there exists a smooth function solving equation (**). Thus the metric is the one we were looking for and the theorem is proved.

Now, an easily stated problem whose solution is not likely to be as short, and has led to a lot of research in the last 2 decades. I read about it for the first time in an interview with Louis Nirenberg, which can be found here.

*Open problem: Given a two dimensional Riemannian manifold, can we embed it isometrically (even just locally) in three dimensional Euclidean space?*

Recall that Gauss curvature solely determines the local geometry of a 2d manifold, so its not surprising that this problem is equivalent to the following problem involving the (…tataaaaa! ) Monge-Ampere equation:

*Open problem restated: Given a function , find (even just locally) a function which solves the equation*

but mind you, this is not your grandpa’s Monge Ampere equation, for it is not an elliptic equation *unless is strictly positive. *For general , it is an equation that varies between hyperbolic or elliptic according to the sign of , so you run into real problems in the set of points where vanishes . If on the other hand is strictly negative, Kazdan says that the equation above is solved using tools from non-linear hyperbolic equations, of which sadly I know nothing. The case is then dealt with the techniques from the last two and the next post, which I should finish in a day or two.