Geometry of CMC surfaces of finite index

Abstract

Given r(0) > 0, I is an element of Nu boolean OR {0}, and K-0, H-0 >= 0, let X be a complete Riemannian 3-manifold with injectivity radius Inj(X) >= r(0) and with the supremum of absolute sectional curvature at most K-0, and let M (sic) X be a complete immersed surface of constant mean curvature H is an element of [0, H-0] and with index at most I. We will obtain geometric estimates for such an M (sic) X as a consequence of the hierarchy structure theorem. The hierarchy structure theorem (Theorem 2.2) will be applied to understand global properties of M (sic) X, especially results related to the area and diameter of M. By item E of Theorem 2.2, the area of such a noncompact M (sic) X is infinite. We will improve this area result by proving the following when M is connected; here, g(M) denotes the genus of the orientable cover of M: (1) There exists C-1 = C-1(I, r(0), K-0, H-0) > 0, such that Area(M) >= C-1(g(M) + 1). (2) There exist C 0 >, G (I) is an element of Nu independent of r(0), K-0, H-0, and also C independent of I such that

if g(M) >= G(I), then Area (M) >= C/(max{1,1/r(0),root K-0, H-0})(2) (g(M) + 1). (3) If the scalar curvature rho of X satisfies 3H(2) +1/2 rho >= c in X for some c > 0, then there exist A, D > 0 depending on c, r(0), K-0, H-0 such that Area(M) <= A and Diameter(M) <= D. Hence, M is compact and, by item 1, g(M) <= A/C-1 - 1.