I put here several common formulas that are needed when we work on stereographic projection. First we try to calculate
We do step by step.
Step 1. A direct computation leads us to
Therefore,
I put here several common formulas that are needed when we work on stereographic projection. First we try to calculate
We do step by step.
Step 1. A direct computation leads us to
Therefore,
This is a classical question: how could we understand the coercivity of the operator ? Given a smooth compact Riemannian manifold of dimension
, the original definition for the coercivity of such an operator is just
for any for some constant
. If
is a function, by taking a positive constant we immediately see that it is necessary
.
Consequently, and in particular if
is constant,
must be positive. If
is not constant, then it is not clear to classify
. In that case, we interpret this terminology in a different manner as follows
It is simple to show that the coercivity of is equivalent to
. The fact that
implies that the coercivity of
is simple just by definition. For the reverse case, it involves a bit of calculus of variation. For interested reader, we refer to a book by O. Druet, E. Hebey, and F. Robert [here].