Quick post: A theorem and an open problem about Gauss curvature

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survey on Prescribing the curvature of a Riemannian manifold, a short book that I recommend strongly (its a bit out of date, but hey!, geometry has advanced quite a bit in the last couple of years!)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 \Omega is a 2-form such that \int_M \Omega = 2\pi \chi(M), then there exists a smooth Riemannian metric g on M such that \Omega = KdA, where K is the Gauss curvature of g and A is the area form of g.

Proof: Pick your favorite metric g_0 on M, we will  prove that the metric we are looking for is in fact conformal to g_0, for any smooth function u, define g_u = e^{2u}g_0. Using the well known formulas for the Gauss curvature and area form of g_u, one arrives at the identity

K_udA_u=K_odA_0-(\Delta_0 u) dA_0

where the sub-indexes 0 and u refer to the object corresponding to the metric g_0 and g_u, in particular, \Delta_0 is the Laplace operator for (M,g_0). Then one wants to pick u such that

K_odA_0-\Delta_0 u dA_0=\Omega

or (\Delta_0 u) dA_0 = K_0dA_0-\Omega

but, using the metric g_0 we can write \Omega = f(x)dA_0 for some f (or equivalently, using the Hodge star operator given by g_0) thus the u we want is the solution of the equation

\Delta_0 u = K_0-f (**)

but since \int_M \Omega =\int_MfdA_0= 2\pi\chi(M)=\int_MK_0dA_0 we have that \int_MK_0-fdA_0=0 thus by standard elliptic theory (or Fredholm theory, etc) there exists a smooth function u solving equation (**). Thus the metric g_u 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 C^\infty function K(x,y), find (even just locally) a function u(x,y) which solves the equation

\frac{u_{xx}u_{yy}-u_{xy}^2}{\left (1+u_x^2+u_y^2 \right)^2} = K(x,y)

but mind you, this is not your grandpa’s Monge Ampere equation, for it is not an elliptic equation unless K is strictly positive. For general K, it is an equation that varies  between hyperbolic or elliptic according to the sign of K, so you run into real problems in the set of points where K vanishes .  If on the other hand K 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 K >0 is then dealt with the techniques from the last two and the next post, which I should finish in a day or two.

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