This question concerns the asymptotic geometry of a sequence of Riemannian metrics on a closed surface whose volume entropies converges to zero.

Let $\Sigma$ be a closed, orientable, connected surface of genus at least $2,$ and suppose $g_{n}$ is a sequence of non-positively curved Riemannian metrics such that the volume entropy satisfies $E(g_{n})\rightarrow 0.$ Here, the volume entropy is the quantity first introduced by Manning \begin{align} E(g_{n}):=\limsup_{R\rightarrow \infty} \frac{\log(\lvert B_{\widetilde{g_{n}}}(x, R)\rvert)}{R}, \end{align} where $\widetilde{g}$ is the pullback metric to the universal cover $\widetilde{\Sigma}$ of $\Sigma$, and $\lvert B_{\widetilde{g_{n}}}(x, R) \rvert$ denotes the area of the ball of radius $R$ in the universal cover $\widetilde{\Sigma}$ where $x\in \widetilde{\Sigma}$ is a base point. It is a basic fact that the volume entropy is independent of the choice of base point, and the limit supremum exists as a limit independent of subsequence.

Next, consider the normalized Riemannian volume measures \begin{align} \mu_{n}:=\frac{dV_{g_{n}}}{Vol(g_{n})} \end{align} which are probability measures on $\Sigma.$

Since the sequence $\mu_{n}$ is a sequence of probability measures on a compact space, there exists a weakly convergent subsequence $\mu_{n_{j}}\rightarrow\mu$.

Now, let $h$ be a background metric on $\Sigma.$ With respect to the volume measure of $h$, the measure $\mu$ has a Lebesgue decomposition into a trio of measures \begin{align} \mu=\mu_{cont}+ \mu_{sing}+\mu_{pp} \end{align} which are the continuous, singular and pure point measures with respect to the background measure induced by $h.$

My question is the following: is it true that the singular part vanishes and the pure point part of the measure $\mu_{pp}$ is supported on a finite set of points. In this regard, this is really a question about the asymptotic distribution of volume for a sequence of Riemannian metrics whose volume entropies are converging to zero.

For a mild bit of context, this question is born from considerations of certain sequences of Riemannian metrics with arise from a sequence of minimal surfaces in symmetric spaces, and I am asking because I want to try to isolate an issue which might have a general answer. This is closely related to the question of whether or not a nearly flat, non-positively curved metric on such a surface must concentrate its negative curvature upon a finite collection of points.

Thank you for any consideration you may give to this question.


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