Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold. Assume that $M$ contains a ray $\gamma : [0, \infty) \to \mathbb{R}$. Let $b_\gamma$ be the associated Busemann function, i.e., $$ b_\gamma(x) = \lim_{t \rightarrow \infty}(d(x, \gamma(t) - t). $$
The Busemann function in general is not smooth. We only know that it is continuous. Let $p \in M$ and let $\tilde{\gamma}$ be another ray starting from $p$ and constructed by using $\gamma$. (In his Riemannian Geometry book, Petersen calls it the asymptote for $\gamma$ from $p$). It is common to introduce the following support function from above for $b_\gamma$ at $p$: $$ b_t(x) \,=\, d(x, \tilde{\gamma}(t)) \,- \,t \,\, + b_\gamma(p). $$ For every $t >0$, this is indeed a support function from above at $p$ for $b_\gamma$ because $b_t$ is smooth around $p$ and
- $b_t(q) = b_\gamma(q)$;
- $b_t (x) \ge b_\gamma(x)$ in a neighbourhood of $p$.
In the literature, I found just this family of support function for $b_\gamma$, but I am dealing with a problem where I need something else. For example, it would be extremely desirable that the following family of functions are a family of support functions: $$ b_t^*(x) = \frac{1}{t} \Big( d(x, \tilde\gamma(t)) - t\Big) + b_\gamma(t). $$ But I can't prove that the second property holds.
Essentially I need a family of support functions where the only term where appears $x$ is of the kind $$ \frac{K}{t^\alpha}d(x, \tilde\gamma(t)), $$ where $K$ is a positive constant and $\alpha \in (0, 1]$.
Any help or suggestion would be extremely appreciated.
