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By Luther Pfahler Eisenhart

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3. Show this inequality. 1 M is conformal to the plane. e. E > 0) we may use the comparison function gE(r) = r(log r)l+<. The curvature of the corresponding metric is now K E(r) and the corresponding test integral is finite. Using a similar argument as above, but with the reversed inequalities we then get g(r) > gE(r) for all r 2: 1. 1 M is conformal to a disc. g. [GroKM] or [BerGM] . 36 Steen Markvorsen ~ ~~~- '--=-~~ ~-:~~~=:=;:. "~ FIGURE 13. A conformal representation of Costa's minimal surface.

G. [CheLY2], Corollary 1 p. 1052 and the generalization in [Ma3], Corollary A p. 481), and then the standard rigidity conclusions follow as ~~. 6. e. g. by using one of the other three Laplace comparison settings alluded to in Section 4 . 7. e. from domains of type Ap,R(p) = DR(p) - Dp(p) ~ not even in constant curvature ambient spaces - unless, of course, the annulus is totally geodesic. 8. What is the mean exit time from a point in the totally geodesic annulus A~',~(p) = B~m(p) - B~,m(p) in Km(b) ?

1 M is conformal to the plane. e. E > 0) we may use the comparison function gE(r) = r(log r)l+<. The curvature of the corresponding metric is now K E(r) and the corresponding test integral is finite. Using a similar argument as above, but with the reversed inequalities we then get g(r) > gE(r) for all r 2: 1. 1 M is conformal to a disc. g. [GroKM] or [BerGM] . 36 Steen Markvorsen ~ ~~~- '--=-~~ ~-:~~~=:=;:. "~ FIGURE 13. A conformal representation of Costa's minimal surface. 4. Let (pm, gP) be a locally conformally fiat Riemannian manifold, i.

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An introduction to differential geometry with use of tensor calculus by Luther Pfahler Eisenhart


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