Verifying the barrier condition for the disk model and stereographic projection

This Mathematica notebook is written to compute the Hessian of functions in spherical geometry. The goal is to bound the Hessian of a particular function to establish certain properties of the fundamental gap.
​
First we set the dimension.
In[]:=
n=2
Out[]=
2
In[]:=
Clear[x1,x2]
Note that it is important not to use the symbols, i, j, k, l, m or n as constants or coordinates in the expressions. The reason is that the first five of those symbols are used as summation or table indices in the calculations done below, and n is the dimension of the space.
In[]:=
xcoord={x1,x2}
Out[]=
{x1,x2}
Since we are working with stereographic projection, we define the potential associated to it. We introduce the parameter R, which indicates the radius of the sphere.
In[]:=
Conformalfactor=4R^4/(R^2+x1^2+x2^2)^2
Out[]=
4
4
R
2
(
2
R
+
2
x1
+
2
x2
)
We now compute the Riemannian metric and inverse metric
In[]:=
Metric:=Metric=ConformalfactorIdentityMatrix[n]
In[]:=
InverseMetric:=InverseMetric=Simplify[Inverse[Metric]]
Christoffel Symbols
We now compute the Christoffel symbols of the associated metric.
In[]:=
ChristoffelLowered:=ChristoffelLowered=Simplify[Table[1/2(D[Metric[[k,i]],xcoord[[j]]]+D[Metric[[k,j]],xcoord[[i]]]-D[Metric[[i,j]],xcoord[[k]]]),​​{k,1,n},{i,1,n},{j,1,n}]]
In[]:=
ChristoffelRaised:=ChristoffelRaised=Simplify[Table[Sum[InverseMetric[[m,k]]ChristoffelLowered[[k,i,j]],{k,1,n}],​​{m,1,n},{i,1,n},{j,1,n}]]
We can display these symbols.
In[]:=
ChristoffelRaised
Out[]=
-
2x1
2
R
+
2
x1
+
2
x2
,-
2x2
2
R
+
2
x1
+
2
x2
,-
2x2
2
R
+
2
x1
+
2
x2
,
2x1
2
R
+
2
x1
+
2
x2
,
2x2
2
R
+
2
x1
+
2
x2
,-
2x1
2
R
+
2
x1
+
2
x2
,-
2x1
2
R
+
2
x1
+
2
x2
,-
2x2
2
R
+
2
x1
+
2
x2

​

Computing the Hessian and its inverse

This is where we define the Hessian and its inverse. To begin, we define a weighting function. This particular function is done so that the composite membrane problem is unitarily equivalent to an eigenvalue problem in hyperbolic space.
In[]:=
weightingfunction=((R^2+x1^2+x2^2)/(R^2(1-x1^2-x2^2)))^2
Out[]=
2
(
2
R
+
2
x1
+
2
x2
)
4
R
2
(1-
2
x1
-
2
x2
)
We now compute the Hessian of this function in terms of spherical geometry. First we compute the second derivatives in the usual way.
In[]:=
EuclideanHess:=EuclideanHess=Simplify[Table[D[weightingfunction,xcoord[[i]],xcoord[[j]]],{i,1,n},{j,1,n}]]
To account for the non-vanishing Christoffel symbols, we have the following terms as well in the Hessian.
In[]:=
ChristoffelCorrection:=ChristoffelCorrection=Simplify[Table[Sum[ChristoffelRaised[[k,i,j]]D[weightingfunction,xcoord[[k]]],{k,1,n}],​​{i,1,n},{j,1,n}]]
In[]:=
SphericalHess:=Simplify[EuclideanHess-ChristoffelCorrection]
In[]:=
MatrixForm[SphericalHess]
Out[]//MatrixForm=
4(1+
2
R
)(
4
x1
+
2
R
(1+5
2
x1
-
2
x2
)+
2
x2
(-1+
2
x2
)+
2
x1
(5+2
2
x2
))
4
R
4
(-1+
2
x1
+
2
x2
)
24
2
(1+
2
R
)
x1x2
4
R
4
(-1+
2
x1
+
2
x2
)
24
2
(1+
2
R
)
x1x2
4
R
4
(-1+
2
x1
+
2
x2
)
-
4(1+
2
R
)(-
4
x1
+
2
R
(-1+
2
x1
-5
2
x2
)+
2
x1
(1-2
2
x2
)-
2
x2
(5+
2
x2
))
4
R
4
(-1+
2
x1
+
2
x2
)
​
What we really need are the eigenvalues of the spherical Hessian.
In[]:=
EigenvaluesofSphericalHess[R_]=Simplify[Eigenvalues[SphericalHess]]
Out[]=
-
4(1+
2
R
)(
2
R
-
2
x1
-
2
x2
)
4
R
3
(-1+
2
x1
+
2
x2
)
,
4(1+
2
R
)(
4
x1
+
2
x2
(5+
2
x2
)+
2
x1
(5+2
2
x2
)+
2
R
(1+5
2
x1
+5
2
x2
))
4
R
4
(-1+
2
x1
+
2
x2
)

​

Finding the values of the parameters

We now want to find the value of R which gives us the largest region to work with in hyperbolic geometry. To this end, we consider the following function
In[]:=
Simplify[Last[EigenvaluesofSphericalHess[R]]-2/R^2weightingfunctionConformalfactor]
Out[]=
(4(-2
2
R
2
(-1+
2
x1
+
2
x2
)
+(1+
2
R
)(
4
x1
+
2
x2
(5+
2
x2
)+
2
x1
(5+2
2
x2
)+
2
R
(1+5
2
x1
+5
2
x2
))))(
4
R
4
(-1+
2
x1
+
2
x2
)
)
We can convert this to a quadratic in r^2 = x1^2+x2^2. The easiest way to do this is to set x2=0, since the potential is rotationally symmetric.
In[]:=
x2=0
Out[]=
0
In[]:=
x1=Sqrt[y]
Out[]=
y
In[]:=
SimplifiedSecondEigenvalueCondition=Simplify[Last[EigenvaluesofSphericalHess[R]]-2/R^2weightingfunctionConformalfactor]
Out[]=
4(y(5+y)+
4
R
(1+5y)-
2
R
(1-14y+
2
y
))
4
R
4
(-1+y)
In[]:=
Numerator[SimplifiedSecondEigenvalueCondition]
Out[]=
4(y(5+y)+
4
R
(1+5y)-
2
R
(1-14y+
2
y
))
In[]:=
Solve[Numerator[SimplifiedSecondEigenvalueCondition]==0,y]
Out[]=
y
-5-14
2
R
-5
4
R
-(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)
,y
-5-14
2
R
-5
4
R
+(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)

In[]:=
Lasty
-5-14
2
R
-5
4
R
-(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)
,y
-5-14
2
R
-5
4
R
+(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)

Out[]=
y
-5-14
2
R
-5
4
R
+(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)

In[]:=
Plot
-5-14
2
R
-5
4
R
+(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)
,{R,0,1}
Out[]=
Now we need to choose R to maximize this quantity.
In[]:=
​​Maximize
-5-14
2
R
-5
4
R
+(1+
2
R
)
25+94
2
R
+25
4
R
2(1-
2
R
)
,R>0,R
Out[]=
5+
7
2
(7-
33
)+
5
16
2
(7-
33
)
-
25+
47
2
(7-
33
)+
25
16
2
(7-
33
)
-
1
4
(7-
33
)
25+
47
2
(7-
33
)+
25
16
2
(7-
33
)
2-1+
1
4
(7-
33
),R
7-
33
2

Here, the first component is the largest possible value of r^2 and the second component is the optimal value of R. We now clear x1 and x2 so that we can plot figures

Plotting figures to ensure the conditions of the theorem hold

From the condition in the theorem, we need that the weighting function is not too convex. In particular, we must have that the following two functions are negative.
​
And just to confirm that the first eigenvalue works as well

Calculating the

Now we use our value for the Euclidean radius to determine the hyperbolic radius.
Here we use the result by Dekster to bound the diameter by the circumradius

Bounds on the weighting function

Later in the proof, we will have to bound the weighting function from above and below.
The lower bound is simple.