I am going through an answer of Robert Bryant to one of my questions from last year; I want to ask about a passing comment he made there.

Just to give a quick summary for context, the answer is an analysis of the geodesics on the 'surface' $\{ (x,y) \in \mathbf{R}^2 \mid x > 0, y > 0 \}$ equipped with the conformal metric $g = (xy)^{2n} (\mathrm{d}x^2 + \mathrm{d}y^2)$. This has Gauss curvature \begin{equation} K = n(x^2 + y^2)/(xy)^{2n} > 0, \end{equation} which diverges to $+\infty$ near the 'boundary' $\{ xy = 0 \}$.

Bryant then makes the comment that this divergence of $K$ 'suggests that nearly all geodesics will avoid going to the singular boundary'.

The comment is ultimately borne out by an ODE analysis, but here I want to ask about its meaning per se. It intuitively makes sense, but I am struggling to even come up with a formal version of the statement.

Question. In a surface $(M^2,g)$ with boundary, where $K \to +\infty$ near $\partial M$, are the geodesics going to $\partial M$ meagre in the Baire sense for example? In general, do 'most' geodesics avoid regions where the curvature is 'large and positive'?