I am motivated by the Lorenz curve used in economics and statistics - a proper multivariate generalization will help researchers (Arnold, Taguchi, etc.) assess multidimensional risk and inequality (important for corporations and small businesses as well). This extension problem is highly important to researchers like me. This problem can be interpreted from a differential geometry standpoint, a multivariate statistical standpoint, and many others. Understanding this extension problem in terms of Lorentzian geometry is highly important to me (I will touch on this at the end).
Prescribe paths $\log x \log y = \log^2 s$ where $s\in (0,1)$ indexes the paths and $x,y \in (0,1).$
Does there exist a real smooth $2$-manifold (aside from the cusps), $M,$ topologically equivalent to the twice pointed sphere (or American football) whose geodesics (spanning from cusps $(0,1,1)$ to $(1,1,0))$ project onto the prescribed paths (above)?
I don't mind if the mapping is not one-to-one but it must be a surjection.
Another phrasing with more requirements is:
Does there exist a real analytic algebraic variety in $(0,1)^3$ which is also a real analytic manifold (by deleting 2 singular points) - whose underlying topological structure is equivalent to the twice pointed sphere - and whose geodesics from $(0,1,1)$ to $(1,0,0)$ correspond to algebraic subvarities (forming the surface) - whose projection to the plane yields the prescribed paths at the top?
Physical relevance:
The paths I gave are clearly a fibration of Minkowski space in (1+1) dimensions, furnished with the appropriate Lorentzian metric, which can be given as $g=\frac{dxdy}{xy}.$ Essentially this is a conformal map $f:\Bbb M^{1,1} \to \zeta^{1,1} $ where one pushes the linear structure(s) from $\Bbb M^{1,1}$ to $\zeta^{1,1}.$ By inspection and from the metric it should be clear that $f:= \exp$ mapping (and this map is indeed linear in this context).
Now to setup the extension problem in this context, we arrange a boundary $B=[0,1]^3$ and a manifold $M=(0,1)^3.$ Assume that slicing $M$ with planes orthogonal to the cubes faces, yields $\zeta^{1,1}.$
Think of a submanifold of $M$ as a balloon with two openings for air at $(0,1,1)$ and $(1,1,0)$ and constant air flow from both openings. The balloon will fill the cube and get arbitrarily close to the boundary of the cube and will start to form creases or sharper edges as it molds into the cube.
This submanifold (surface) being fixed with a certain interior volume is basically what I am asking about above - asking if it can be regarded as a real analytic algebraic variety as well as manifold. Also I have another requirement about the projection of certain curves (that span both ends of the balloon) assumed to be geodesics from the manifold viewpoint.
The main question here is can we equip the $3$-manifold $M$ with a Lorentzian metric...