See http://link.springer.com/article/10.1007%2FBF02395060 Lemma 12.2 (the $\gamma_i$ terms are defined on page 167, also see $\S2$ for the curvature notations).

An alternative "do-it yourself" approach to what you want might be to consider the expansion of the metric in normal coordinates, e.g. Riemann's formula for the metric in a normal neighborhood, which is not that hard to prove. Then plug this into the coordinate expression for the Laplacian and find a power series for the Laplacian. Changing into polar coordinates should then presumably give you what you want.

EDIT: Here are is the formula written out.

For $m\in M$, the mean curvature at the point $\exp_m(ru)$ (for $r >0$ and $u \in T_mM$ a unit vector) is given by $$ \frac{n-1}{r} + \alpha_1r+\alpha_2r^2+O(r^3) $$ where \begin{align*} \alpha_1 & = -\frac13 Ric|_m(u,u)\\ \alpha_2 & = -\frac14(\nabla Ric)|_m (u,u,u). \end{align*}

See http://link.springer.com/article/10.1007%2FBF02395060 Lemma 12.2 (the $\gamma_i$ terms are defined on page 167, also see $\S2$ for the curvature notations).

An alternative "do-it yourself" approach to what you want might be to consider the expansion of the metric in normal coordinates, e.g. Riemann's formula for the metric in a normal neighborhood, which is not that hard to prove. Then plug this into the coordinate expression for the Laplacian and find a power series for the Laplacian. Changing into polar coordinates should then presumably give you what you want.

EDIT: Here are is the formula written out.

For $m\in M$, the mean curvature at the point $\exp_m(ru)$ (for $r >0$ and $u \in T_mM$ a unit vector) is given by $$ \frac{n-1}{r} + \alpha_1r+\alpha_2r^2+O(r^3) $$ where \begin{align*} \alpha_1 & = -\frac13 Ric|_m(u,u)\\ \alpha_2 & = -\frac14(\nabla Ric)|_m (u,u,u). \end{align*}

See http://link.springer.com/article/10.1007%2FBF02395060 Lemma 12.2 (the $\gamma_i$ terms are defined on page 167, also see $\S2$ for the curvature notations).

An alternative "do-it yourself" approach to what you want might be to consider the expansion of the metric in normal coordinates, e.g. Riemann's formula for the metric in a normal neighborhood, which is not that hard to prove. Then plug this into the coordinate expression for the Laplacian and find a power series for the Laplacian. Changing into polar coordinates should then presumably give you what you want.

See http://link.springer.com/article/10.1007%2FBF02395060 Lemma 12.2 (the $\gamma_i$ terms are defined on page 167, also see $\S2$ for the curvature notations).

An alternative "do-it yourself" approach to what you want might be to consider the expansion of the metric in normal coordinates, e.g. Riemann's formula for the metric in a normal neighborhood, which is not that hard to prove. Then plug this into the coordinate expression for the Laplacian and find a power series for the Laplacian. Changing into polar coordinates should then presumably give you what you want.

EDIT: Here are is the formula written out.

For $m\in M$, the mean curvature at the point $\exp_m(ru)$ (for $r >0$ and $u \in T_mM$ a unit vector) is given by $$ \frac{n-1}{r} + \alpha_1r+\alpha_2r^2+O(r^3) $$ where \begin{align*} \alpha_1 & = -\frac13 Ric|_m(u,u)\\ \alpha_2 & = -\frac14(\nabla Ric)|_m (u,u,u). \end{align*}