1. Introduction
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 .
2. Proof
Consider and defined on which satisfies that is supported on .
Set , then
 G is nonnegative 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 CauchySchwarz 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 :
which is:
which means:
 Consider , where , then

Take it into (15),
Thus
Then
Because is the maximum point, then
restrict this to , then we have:
therefore
i.e,
2.1. Remark
 This estimate is sharp, when we come to the example of linear functions.
 We can replace the cutoff function by other cutoff functions.
3. Application
 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.
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