Index estimates for planar domains with Robin boundary condition

In this note, we study the index of the Laplace operator on planar domains $\mathrm{\Omega }$ with compact smooth boundary, allowing both the compact and noncompact cases, under a natural Robin boundary condition involving the geodesic curvature of $\mathrm{\partial }\mathrm{\Omega }$. We obtain lower bounds for the index in terms of the number of boundary components. Our approach combines conformal and spectral techniques with the topology of the domain, encoded in the space of $L^2$ harmonic vector fields tangent to the boundary. The result is motivated by index estimates for free boundary minimal surfaces and provides an intrinsic counterpart to a borderline geometric situation arising in that setting.