### 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survey on Prescribing the curvature of a Riemannian manifold, a short book that I recommend strongly (its a bit out of date, but hey!, geometry has advanced quite a bit in the last couple of years!)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.