Relation between function and Laplacian in Riemannain manifold

Question

Let $f$ and $k$ be two real valued function on the Riemannian manifold $(M,g)$ such that for each $p\in M$, $$k(p)\geq \frac{\partial f}{\partial x_i}|_p.$$ Then wthat is the relation between Laplacian $\Delta f$ and $k$?
I have tried as follows
Since $grad(f)=g^{ij}\frac{\partial f}{\partial x_j}\partial _i$, hence $grad(f)|p\leq k(p)g^{ij}\partial _i|p$. Then putting the value of $grad(f)$ in Laplacian we get $$\Delta(f)(p)=g_p(\nabla_j k(p)g^{ij}\partial _i,\partial_j).$$ After that I can not go further. Please anyone help me. Thank you

Answer 1

It is not possible to bound the Laplacian with a function of the sole $k$. For example in $\mathbb{R}$ with the standard metric you can have bounded derivatives with unbounded second derivatives.

On the other hand we know that in the particular chart you have used $$ \Delta f = g^{i j} \frac{\partial^2 f}{\partial x^i \partial x^j} -g^{i j} \Gamma^k_{i j}\frac {\partial f}{\partial x^k} \leq g^{i j} \frac{\partial^2 f}{\partial x^i \partial x^j} + |k| \sum_{i,j,l}|g^{ij} \Gamma^l_{i j}|$$