Advanced Nonlinear Studies (May 2023)
Geometry of CMC surfaces of finite index
Abstract
Given r0>0{r}_{0}\gt 0, I∈N∪{0}I\in {\mathbb{N}}\cup \left\{0\right\}, and K0,H0≥0{K}_{0},{H}_{0}\ge 0, let XX be a complete Riemannian 3-manifold with injectivity radius Inj(X)≥r0\hspace{0.1em}\text{Inj}\hspace{0.1em}\left(X)\ge {r}_{0} and with the supremum of absolute sectional curvature at most K0{K}_{0}, and let M↬XM\hspace{0.33em}\looparrowright \hspace{0.33em}X be a complete immersed surface of constant mean curvature H∈[0,H0]H\in \left[0,{H}_{0}] and with index at most II. We will obtain geometric estimates for such an M↬XM\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↬XM\hspace{0.33em}\looparrowright \hspace{0.33em}X, especially results related to the area and diameter of MM. By item E of Theorem 2.2, the area of such a noncompact M↬XM\hspace{0.33em}\looparrowright \hspace{0.33em}X is infinite. We will improve this area result by proving the following when MM is connected; here, g(M)g\left(M) denotes the genus of the orientable cover of MM: (1)There exists C1=C1(I,r0,K0,H0)>0{C}_{1}={C}_{1}\left(I,{r}_{0},{K}_{0},{H}_{0})\gt 0, such that Area(M)≥C1(g(M)+1){\rm{Area}}\left(M)\ge {C}_{1}\left(g\left(M)+1).(2)There exist C>0C\gt 0, G(I)∈NG\left(I)\in {\mathbb{N}} independent of r0,K0,H0{r}_{0},{K}_{0},{H}_{0}, and also CC independent of II such that if g(M)≥G(I)g\left(M)\ge G\left(I), then Area(M)≥C(max1,1r0,K0,H0)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 XX satisfies 3H2+12ρ≥c3{H}^{2}+\frac{1}{2}\rho \ge c in XX for some c>0c\gt 0, then there exist A,D>0A,D\gt 0 depending on c,I,r0,K0,H0c,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, MM is compact and, by item 1, g(M)≤A/C1−1g\left(M)\le A\hspace{0.1em}\text{/}\hspace{0.1em}{C}_{1}-1.
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