A question about an estimate

Question

Let $ f:M \rightarrow N $ be a minimal immersion where $ M $ is a compact two dimensional manifold and $ N $ is a three dimensional manifold. Let $ |A|^2 $ be the square of its second fundamental form. Let $ c $ be an upper bound of the norm of the riemannian curvature tensor of $ N $ ( $ |R|\leq c $ where $R$ is the riemannian curvature tensor of $ N $ ). Let $ \int_M |A|^2 dv \leq c $ and $ \int_M |A|^{\frac{5}{2}} dv \leq c $ . I want to prove that $ \int_M |K|^{\frac{5}{4}} dv \leq c' $ where c' depending only on $ c$ and $ K $ is the sectional curvature of $ M$. This problem is related to an estimate in the article 'Estimates for stable minimal surfaces in three dimensional manifolds' Schoen. I have been thinking that, since the immersion is minimal, $ \frac{1}{2}|A|^2=R_{1221}-K $ where $ R_{1221} $ is the riemannian curvature tensor of $N$ in the two directions tangent to $ M $. This equality could be usefull but i don't know as apply it.Thank you

Answer 1

Since the immersion is minimal, as you noted by the Gauss-Codazzi equations

$$ \frac12 |A|^2 = R_{1221} - K $$

re-organizing the terms you have

$$ K = R_{1221} - \frac12 |A|^2 $$

So raising to a power and integrating you get

$$ \int_M |K|^{5/4} \mathrm{d}v \leq 2\left( \int_M |R_{1212}|^{5/4} \mathrm{d}v + \frac1{4\sqrt{2}} \int_M |A|^{5/2} \mathrm{d}v \right) $$

from judicial application of the triangle inequality: $$|X+Y|^p \leq ||X| + |Y||^p \leq 2 \max(|X|,|Y|)^p \leq 2 |X|^p + 2|Y|^p~.$$ Using the $L^\infty$ estimate for $R$ as well as the estimate for $\|A\|_{5/2}$ we get

$$ \int_M |K|^{\frac54} \mathrm{d}v \leq 2 \left( c^{\frac54} \int_M \mathrm{d}v + \frac1{4\sqrt{2}} c\right) $$