Complete Properly Embedded Minimal Surfaces in R3

In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface 2 ⊂ R3 with finite topology. The main result is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature. In this short paper, we apply estimates and ideas from [CM4] to study the ends of a properly embedded complete minimal surface 2 ⊂ R3 with finite topology. For such surfaces, each end has a representative E that is a properly embedded minimal annulus. If E has finite total curvature, it is asymptotic to either a plane or half of a catenoid. On the other hand, the helicoid provides the only known example of an end with finite topology and infinite total curvature. Clearly, no representative for the end of the helicoid can be disjoint from an end of a plane or catenoid. The main result of this paper is that any complete properly embedded minimal annulus that lies above a sufficiently narrow downward sloping cone must have finite total curvature. This is closely related to a result of P. Collin [Co] described below. In [HoMe], D. Hoffman and W. Meeks proved that at most two ends of can have infinite total curvature. Further, they conjectured that if as above has at least two ends, then it must have finite total curvature (the so-called finite total curvature conjecture; see [Me]). If has at least two ends, then there is either an end of a plane or a catenoid disjoint from (see [HoMe, Lemma 5]). Therefore, to prove the finite total curvature conjecture, it suffices to show that a properly embedded minimal annular end E that lies above the bottom half of a catenoid has finite total curvature. In this direction, Meeks and H. Rosenberg [MeR] showed that if has at least two ends, then is conformally equivalent to a compact Riemann surface with finitely DUKE MATHEMATICAL JOURNAL Vol. 107, No. 2, c © 2001 Received 23 February 1999. Revision received 26 June 2000. 2000 Mathematics Subject Classification. Primary 53C21, 53C43. Colding’s work partially supported by National Science Foundation grant number DMS-9803253 and an Alfred P. Sloan Research Fellowship. Minicozzi’s work supported by National Science Foundation grant number DMS-9803144 and an Alfred P. Sloan Research Fellowship.