Topics and some Solutions in Minimal Surfaces, Geodesics and Networks as they pertain to minimizing cost and tension in economic equilibrium..

1. Richard Anthony Baum April 8, 2017
Lecturer, Department of Economics
Allan Hancock College
Economic Networks and Minimal Surface Economies:
Topics and A Solution
(1) Provide an economic interpretation for the components of the metric for the map
( , , ( , ))X u v f u v where u and v are denominate signals and ( , )f u v is a denominate premium
reduction or alternatively for the map ( , , ( , ))X u v u v where u and v are as before, and now z is a
denominate fixed premium and ( , ) ( , ) 0u v z f u v    is a denominate isoprofit function. Recall
components of the metric are:
2
, 1u u uE X X f  
,u v u vF X X f f  and
2
, 1v v vG X X f   .
(2) Show excess demand evolution to equilibrium may be modeled as a Ricci flow of the
metric. Show on a manifold the contraction mapping theorem may be modeled as contraction of the mean
curvature flow. Explain how solitons are fixed points of a mapping of the equilibrium manifold under the
Ricci flow evolution of the metric.
(3) Thesis Topic: State and prove the theorem every harmonic map which is a saddle
minimal surface is a minimax solution at every point in some choice of co-ordinates. Prove the Nash

2. equilibria solution space to a smooth public differential game with public preimage strategies (u , v) and
isothermal payoff vector 1 2 2( ( , ), ( , ), , ( , ))nX x u v x u v x u v , n , is a minimal submanifold.
(4) Prove a specialization of the converse of this theorem, namely: Every Nash Equilibrium is
a harmonic map for a discrete set of points. To prove this, take the unit vector corresponding to
( , , ( , )), 1, ,2 ,iu v X u v i n and map it to the unit sphere. Such points on the unit sphere are called the
discrete Gauss map of such unit vectors. The discrete Gauss map of all unit vectors
( , , ( , ))
( , , ( , )
i
i
u v X u v
u v X u v
is
trivially harmonic as the Laplacian of a point vanishes (is identically zero). Hence, discrete Nash
Equilibria are trivially harmonic.
This yields a discrete network of points separated by arc lengths joining vertices or nodes of our
network. We solve our network for critical points of the arc length functional or what is the same thing,
critical points of the mathematical energy functional, such critical points typically yielding geodesics.
When transport of information in our network is along geodesics, and assuming a cost functional
increases with arc length, we may be seen to be minimizing the geodesic flow. A geodesic is the one
dimensional analog to a two dimensional minimal surface. That is, in our network, we are minimizing arc
lengths that costly information must travel, whereas in the smooth case of a two-dimensional minimal
surface, we are minimizing the area costly information must span.
In the case of a smooth minimal surface which may be seen as the completion of discrete network,
when the geodesics in our domain map under projection to geodesics in our target or surface, our smooth
surface is totally geodesic. A smooth totally geodesic surface is a minimal surface which minimizes
energy or cost, so that we pay minimal cost for harmonicity or minimizing tension on our surface or in
our economy. Minimizing tension in our surface, our surface is perfectly elastic, so when a force arises
which deforms our surface, our surface springs elastically back to its original state once the force is
removed. This means we minimize tension and achieve harmonicity in economic equilibrium.