2022/12/29 by Meeks, William H., Perez, Joaquin
#49Q05 #53A10 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.14428
Given r0>0, I∈ ℕ∪ \0\ and K0,H0≥ 0, let X be a complete Riemannian 3-manifold with injectivity radius Inj(X)≥ r0 and with the supremum of absolute sectional curvature at most K0, and let M\looparrowright X be a complete immersed surface of constant mean curvature H∈ [0,H0] and with index at most I. We will obtain geometric estimates for such an M\looparrowright X as a consequence of the Hierarchy Structure Theorem in [9]. The Hierarchy Structure Theorem (see Theorem 2.2 below) will be applied to understand global properties of M\looparrowright X, especially results related to the area and diameter of M. By item E of Theorem 2.2, the area of such a non-compact M\looparrowright 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 C1=C1(I,r0,K0,H0)>0 such that Area(M)≥ C1(g(M)+1). 2. There exists C>0,G(I)∈ ℕ independent of r0,K0,H0 and also C independent of I such that if g(M)≥ G(I), then Area(M)≥ \fracC(max\1,(1)/(r0),√(K0), H0\)2(g(M)+1). 3. If the scalar curvature ρ of X satisfies 3H2+(1)/(2)ρ≥ c in X for some c>0, then there exist A,D>0 depending on c,I,r0,K0,H0 such that Area(M)≤ A and Diameter(M)≤ D. Hence, M is compact and, by item 1, g(M)≤ A/C -1.