Positive Harmonic Functions on Complete Manifolds with Non-Negative Curvature Outside a Compact Set

1987 ◽  
Vol 125 (1) ◽  
pp. 171 ◽  
Author(s):  
Peter Li ◽  
Luen-Fai Tam
Keyword(s):  
Harmonic Functions ◽  
Negative Curvature ◽  
Compact Set ◽  
Complete Manifolds ◽  
Positive Harmonic Functions

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

Sign in / Sign up

Export Citation Format

Share Document