J an 2 00 5 Minimal surfaces with the area growth of two planes ; the case of infinite symmetry

Consider a properly immersed minimal surface M in R with area A(r) in balls B(r) of radius r centered at the origin. By the monotonicity formula, the function A(r) = A(r) r2 is monotonically increasing. We say that M has area growth constant A(M) ∈ (0,∞], if A(M) = limr→∞A(r). Note that under a rigid motion or homothety M ′ of M , the number A(M) = A(M ′), and so, A(M) ≥ π, which is the area growth constant of a plane. We say that M has quadratic area growth, if A(M) < ∞. Basic results in geometric measure theory imply that for any M with quadratic area growth and for any sequence of positive numbers ti → 0, the sequence homothetic shrinkings M(i) = tiM ofM contains a subsequence that converges on compact subsets of R 3 to a limit partially supported by NSF grant DMS-0405836. partially supported by NSF grants DMS-9971563 and DMS-0139887. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF.