Let $(M^n,g)$ be a smooth complete Riemannian manifold. Let $p\in M$ be a point. Recall that the cut locus of $p$ is the set of vectors $v$ in the tangent space $T_pM$ such that $\exp(t v)$ is a minimizing geodesic for any $t\in [0,1]$, but not for $t\in [0,1+\varepsilon )$ for any $\varepsilon >0$.
Question. Does the cut locus have Lebesgue measure 0 in $T_pM$? If yes, does it have Hausdorff dimension at most $n-1$?
If the above questions have negative answers, one may ask the same questions about the exponential image of the cut locus.
Best Answer
The key fact is that the cut time $t_c : UM \to \mathbb{R}$, defined on the unit tangent bundle $UM$ of a complete, $n$-dimensional Riemannian manifold, is locally Lipschitz continuous around all $v \in UM$ such that $t_c(v) < +\infty$. Hence the tangential cut locus at $p \in M$, that is $$ \tilde{C}_p = \{t_c(v)v\mid v \in UM,\quad t_c(v) < +\infty\} \subset T_q M, $$ either is empty, or it has Hausdorff dimension exactly $n-1$ (being the graph of a locally Lipschitz function). The exponential map being smooth, it cannot increase the Hausdorff dimension, hence $\dim(C_p) = \dim(\exp_p(\tilde{C}_p)) \leq n-1$.
All of this is proved here by Itoh and Tanaka.