Assume that is a noncompact Riemann manifold and harmonic and positive if the Ricci curvature on has a lower bound for some , then
We are going to prove the case , i.e, . Set .
Consider and defined on which satisfies that is supported on .
Set , then
- G is non-negative on and on . Thus G attains its maximum in , say at . Then
- Consider function ,we have
- Choose orthogonal frame , s.t.
- , whenever .
- , which means .
- Because of the special frame, we get that
- Consider the equations above, then
Here we used Cauchy-Schwarz Inequality and (3).
- Because , we obtain:
- Thus by (4),
- Substitute in the last inequality:
- Before we continue:
- Therefore we multiply (8) with . Then by (9),
- Let’s put this at the point . We have:
- Then at ,
- Mutiply (11)with :
- Consider , where , then
Take it into (15),
Because is the maximum point, then
restrict this to , then we have:
- This estimate is sharp, when we come to the example of linear functions.
- We can replace the cutoff function by other cutoff functions.
- If is positive in the whole space, then is a constant.
- What if when the domain is ?
- When , then is a constant.
- Prove Harnack Inequality.