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Let $u$ be a weak solution of the free boundary problem $$\mathcal Lu=\lambda_0 \mathcal H^1\lfloor\partial\{u>0\}, u\ge 0,$$ where $\mathcal Lu={\text{div}}(g(\nabla u)\nabla u)$ is a quasilinear elliptic operator and$g(\xi)$ is a given function of $\xi$ satisfying some structural conditions. Weprove that the free boundary $\partial\{ u>0\}$ is continuously differentiablein $\mathbb R^2$, provided that $\partial\{ u>0\}$ has locally finiteconnectivity. Moreover, we show that the free boundaries of weak solutions withfinite $\it{Morse \ index}$ must have finite connectivity. The weak solutionsare locally Lipschitz continuous and non-degenerate stationary points of theAlt-Caffarelli type functional $J[u]=\int_{\Omega}F(\nabla u)+Q^2\chi_{\{u>0\}}$.  The full regularity of the free boundary is not fully understood even for the{\it minimizers} of $J[u]$ in the simplest case $g(\xi)=|\xi|^{p-2}, p>1$,partly because the methods from the classical case $p=2$ cannot be generalizedto the full range of $p$. Our method, however, is very geometric and works evenfor the $ stationary\ points$ of the functional $J[u]$ for a large class ofnonlinearities $F$.

A geometric approach to regularity for nonlinear free boundary problems with finite Morse index