Let us have Riemannian manifold $M$ with boundary $N.$ Let $F$ be an immersion, such that $F:N\to M$ and $B$ be a second fundamental form on $N$ relative to $F.$ And let $f$ be a function on $N.$

How we can calculate integral

$$ \int_{N}B(grad(f),grad(f))\Omega, $$

where $\Omega$ is a volume form on $N?$

Is there exist any estimates for this integral?



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