Geometry of CMC surfaces of finite index

Abstract

Abstract Given r 0 > 0 {r}_{0}\gt 0 , I ∈ N ∪ { 0 } I\in {\mathbb{N}}\cup \left\{0\right\} , and K 0 , H 0 ≥ 0 {K}_{0},{H}_{0}\ge 0 , let X X be a complete Riemannian 3-manifold with injectivity radius Inj ( X ) ≥ r 0 \hspace{0.1em}\text{Inj}\hspace{0.1em}\left(X)\ge {r}_{0} and with the supremum of absolute sectional curvature at most K 0 {K}_{0} , and let M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] H\in \left[0,{H}_{0}] and with index at most I I . We will obtain geometric estimates for such an M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}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 ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X , especially results related to the area and diameter of M M . By item E of Theorem 2.2, the area of such a noncompact M ↬ X M\hspace{0.33em}\looparrowright \hspace{0.33em}X is infinite. We will improve this area result by proving the following when M M is connected; here, g ( M ) g\left(M) denotes the genus of the orientable cover of M M : (1) There exists C 1 = C 1 ( I , r 0 , K 0 , H 0 ) > 0 {C}_{1}={C}_{1}\left(I,{r}_{0},{K}_{0},{H}_{0})\gt 0 , such that Area ( M ) ≥ C 1 ( g ( M ) + 1 ) {\rm{Area}}\left(M)\ge {C}_{1}\left(g\left(M)+1) . (2) There exist C > 0 C\gt 0 , G ( I ) ∈ N G\left(I)\in {\mathbb{N}} independent of r 0 , K 0 , H 0 {r}_{0},{K}_{0},{H}_{0} , and also C C independent of I I such that if g ( M ) ≥ G ( I ) g\left(M)\ge G\left(I) , then Area ( M ) ≥ C ( max 1 , 1 r 0 , K 0 , H 0 ) 2 ( g ( M ) + 1 ) {\rm{Area}}\left(M)\ge \frac{C}{{\left(\max \left\{1,\frac{1}{{r}_{0}},\sqrt{{K}_{0}},{H}_{0}\right\}\right)}^{2}}\left(g\left(M)+1) . (3) If the scalar curvature ρ \rho of X X satisfies 3 H 2 + 1 2 ρ ≥ c 3{H}^{2}+\frac{1}{2}\rho \ge c in X X for some c > 0 c\gt 0 , then there exist A , D > 0 A,D\gt 0 depending on c , I , r 0 , K 0 , H 0 c,I,{r}_{0},{K}_{0},{H}_{0} such that Area ( M ) ≤ A {\rm{Area}}\left(M)\le A and Diameter ( M ) ≤ D {\rm{Diameter}}\left(M)\le D . Hence, M M is compact and, by item 1, g ( M ) ≤ A / C 1 − 1 g\left(M)\le A\hspace{0.1em}\text{/}\hspace{0.1em}{C}_{1}-1 .