(Regularity of teardrops) Let $u:\Sigma\to X$ be a teardrop curve with a unique output $q$ and no input ends. Assume $v_{1},\ldots v_{n-1}$ are in the kernel of the linearized Cauchy-Riemann operator on $W^{1,2;\mu=\pi}$, such that $\Omega(\cdot,v_{1},\ldots v_{n-1})$ does not vanish identically as a 1-form on $\Sigma$. Then $\deg q\geq 2$. When the equality is achieved, the teardrop curve is an immersion up to the boundary with minimal vanishing at the corner, and the kernel of the ordinary Cauchy-Riemann operator is spanned as a real vector space by the Möbius vector fields on $\Sigma$ fixing the $q$ corner, and the cokernel vanishes.