Laplacian vanishing theorem for a quantized singular Liouville equation

In this article we establish a vanishing theorem for a singular Liouville equation with a quantized singular source. If a blowup sequence tends to infinity near the quantized singular source and the blowup solutions violate the spherical Harnack inequality around the singular source (non-simple blow-ups), the Laplacian of the coefficient function must tend to zero. This seems to be the first second order estimates for a Liouville equation with a quantized source and non-simple blowups. This result as well as the key ideas of the proof will be useful for various applications.