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There are three invariant curvature functions defined on any smooth space-like 2-surfaces in four-dimensional Minkowski space. (If the surface lies in a Euclidean hyperplane then the functions agree with H2, K2, and (H2 K)2 . For each of these functions we show that there exists a space-like immersion of any oriented compact (or noncompact complete) surface with associated total curvature arbitrarily small. A two dimensional oriented surface M in four dimensional Minkowski space which inherits a positive definite metric is called space-like. Such a surface carries three functions H, K, and U that are determined by the first and second fundamental forms. (If the surface lies in a Euclidean hyperplane then these functions reduce to H , K 2, and (H 2-K 2) respectively.) It is natural to ask for the infimum of the respective total curvatures H dA, fM K 1/2 dA and fM U'12 dA over the set of embedded compact space-like surfaces. In this paper we show that there exists an embedded smooth compact orientable surface Mg of genus g > 0 with fMg IHI dA arbitrarily small (Theorem 5). In the process we relate H to the split mean curvatures in the manner of reference [2]. We then show that there exist compact (and complete noncompact) embedded orientable surfaces of arbitrary genus with either K or U identically zero, (Theorems 6 and 7). The author would like to thank P. Ehrlich, R. Howard, T. K. Milnor and R. Penrose for inspiration. Throughout this paper M4 will denote Minkowski space, the real four dimensional vector space equipped with bilinear form ( , ) of type (3, 1), (i.e., the normal form has 3 plus signs and 1 minus sign). We will assume that M4 is oriented and time oriented (i.e., a 4-volume form dV, and future time-like vector field Fut, have been chosen.) We will write LC = {v e M4 (v, v) = 0} to denote the light cone in Minkowski space. A space-like surface in M4 is Received by the editors February 24. 1989 and, in revised form, September 1 1, 1989. 1980 Mathcmatics Subjyct Classification (1985 Revision). Primary 53C50; Secondary 53B30, 53C42. Research partially supported by NSF grant DMS 88-03585. ? 1990 American Mathematical Society 0002-9939/90 $1 00 + $ 25 per page

Lower bounds for the extrinsic total curvatures of a space-like codimension 2 surface in Minkowski space