Is the following claim true ?
Given a smooth and bounded $u:M\to\mathbb{R}$ , where $(M,g)$ is a Riemannian manifold , then there exists a harmonic extension $\tilde{u}:M\times\mathbb{R}^+\to\mathbb{R}$ as the unique bounded function satisfying $$\begin{cases}\Delta_{\bar{g}}\tilde{u}=0 \ \ \text{in} \ \ M\times\mathbb{R}^+\\ \tilde{u}=u \ \ \ \ \ \ \ \text{on} \ \ M\times\{0\}\end{cases}$$ where $\bar{g}=g+|dx|^2$ is the product metric on $M\times\mathbb{R}^+$ .
A hint was given to use the maximum principles . But I am not too familiar with that in manifold setting . Any help is appreciated .