TITLE: Curve shortening flow with boundary
QUESTION [5 upvotes]: Let $(M^2,g)$ be a 2-dimensional complete Riemannian manifold (e.g. $(\mathbb{R}^2,\delta_{ij})$) and $p,q\in M$ two points with $p\neq q$. Let $\gamma:I\to M$ be a smooth embedded curve starting at $p$ and ending at $q$.
When does the curve shortening flow starting at $\gamma$ converge to a geodesic segment joining $p$ and $q$, i.e. the "shortest" curve joining the two points?
I am aware of Grayson's result that an embedded closed curve in a 2-manifold either shrinks to a round point or converges to a geodesic. 
What results are known for curve shortening flow for line segments? Is the flow even well defined?

REPLY [2 votes]: Two references that might help: 
(1) "Midpoint geodesic polygon / Birkhoff curve shortening": Bowditch: "The convergence of the Birkhoff process seems to be an open question for Riemanninan 2-manifolds."

          


(2) The paper below explores what they call the disk flow, which
replaces arcs of a curve with geodesic segments:

Hass, Joel, and Peter Scott. "Shortening curves on surfaces." Topology 33, no. 1 (1994): 25-43. (PDF download from Semantic Scholar.)
{"set_name": "stack_exchange", "score": 5, "question_id": 2859554}