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I have the following problem:

Let M be a compact connected riemannian manifold with $\partial M=\emptyset$ and $f \in C^{\infty}(M)$ and $\Delta f\geq0$. Show that f is constant.

To show that f ist constant you can also show that $grad f=0$. Could you use the Green's identities to show that?. How can you use them?

Thanks in advance.

Tobi92sr
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  • What's a "coherently riemannian manifold"? – Jack Lee Jun 29 '17 at 14:33
  • You can apply Green's first identity or just the divergence theorem (pretty much the same thing with the appropriate choice of the fields involved): $\int_M \Delta f = \int_{\partial M} \cdots = 0$ since the boundary is empty. Then apply the conditions on $f$ to get $\Delta f = 0$. From this you can deduce $\nabla f = 0$ for example by applying Green's first identity one more time to a combination like $\int_M (\nabla f)^2 = \int_M f\Delta f + (\nabla f)^2 = \int_{\partial M} \cdots$. – Winther Jun 29 '17 at 14:43

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