Singularities of the solution to a Monge–Ampère equation on the boundary of the 3-simplex

Abstract We show that the metric defined by the solution to the tropical Monge–Ampère equation, as defined by Hultgren, Mazzon, and the first two authors, on the boundary of the 3-simplex is asymptotic to the Gross–Wilson metric on $$S^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> near each of the 6 singular points. We deduce in addition that the solution is not $$C^{1,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> across the singular points. Compared to previous works, our starting point is the real Monge–Ampère equation, as opposed to the complex structure.