Conjugate points on cut locus

Intuitively, this seems plausible if $M$ is complete. Let $d:\partial M \to [0,\infty]$ be the distance function to the cut locus. Then $d$ is continuous (by completeness), and $graph(d) \subset \partial M\times [0,\infty]$ is a copy of $\partial M$. There is an open subset $V\subset \partial M$ on which $d$ is finite.

Then $M$ is obtained by taking $U=\{ (x,t)\subset \partial M\times [0,\infty) | t\leq d(x)\}$ and taking the quotient of $graph(d_{|V})$ by the identification map to the cut locus.

If the multiplicity $\geq 2$ points are not dense in the cut locus, then there is an open subset in the cut locus of multiplicity one. This should pull back to an open subset of $graph(d_{|V})$, which maps homeomorophically under the identification map. But an open ball subset of $graph(d_{|V})$ mapping to the cut locus with multiplicity one will have a collar neighborhood homeomorphic to a half-space, which maps homeomorphically to $M$. However, the boundary of the half-space maps to the cut locus, not to $\partial M$, which is a contradiction to the fact that $M$ is a manifold near the cut locus.

Let me sketch another proof. It use analysis instead of topology. Namely we use existence of flow for continuous vector fields (uniqueness requires more, but we will not use it).

Denote by $K$ the subset in the interior of $M$ with multiplicity 1 and let $U\subset K$ be its interior. We need to show that $U$ does not intersect the cut locus.

Consider the function $f= \mathrm{dist}_{∂M}$. Note that $K$ is the set of points where $f$ is differentiable, $|\nabla_x\,f|=1$ for any $x\in K$ and $\nabla f$ is a continuous vector field in $U$. In particular $f$-gradient flow exist in $U$ (in fact, it is unique, but we do not need it).

Fix $x \in U$. Moving $x$ along the $f$-gradient flow, we get a curve $\gamma$ such that $(f\circ\gamma)'\equiv 1$. In other words, $\gamma$ is an extension of the minimizing geodesic from $∂M$ to $x$. Hence $x$ does not lie on the cut locus.

Since $x\in U$ is arbitrary, the statement follows.

I've got a letter from Stephanie Alexander with a complete answer. Let me summarize it here.

The first proof is given in 4.8 of "Schnittort und konvexe Mengen..." by Hermann Karcher (1968). (The formulation is slightly weaker, but from the proof proves our statement follows; the idea is the same as in the answer of Ian Agol.)

An other proof was given in "Decomposition of cut loci" by Richard Bishop (1977).

Latter it appears as Lemma 2 in "Distance function and cut loci..." by Franz-Erich Wolter (1979). The proof is basically the same as Karcher's.