Index estimates for complete noncompact free boundary minimal surfaces

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be an unbounded domain with smooth boundary and let $\mathrm{\Sigma }$ be a complete, noncompact, orientable, immersed free boundary minimal surface in $\mathrm{\Omega }$, with compact boundary $\mathrm{\partial }\mathrm{\Sigma }\subset \mathrm{\partial }\mathrm{\Omega }$ and finite Morse index. We prove that if the mean curvature of $\mathrm{\partial }\mathrm{\Omega }$ satisfies $H_{\mathrm{\partial }\mathrm{\Omega }}\ge 0$ along $\mathrm{\partial }\mathrm{\Sigma }$ and $H_{\mathrm{\partial }\mathrm{\Omega }}>0$ at some point of $\mathrm{\partial }\mathrm{\Sigma }$, then $$\text{Ind}(\mathrm{\Sigma })\ge \frac{1}{3}(2g+k+2\sum _{j=1}^r(d_j+1)-2),$$ where $g$ is the genus of $\mathrm{\Sigma }$, $k$ is the number of connected components of $\mathrm{\partial }\mathrm{\Sigma }$, $r$ is the number of ends of $\mathrm{\Sigma }$ and $d_1,\dots ,d_r$ are their respective multiplicities. When $H_{\mathrm{\partial }\mathrm{\Omega }}$ is only assumed to be nonnegative we obtain $\text{Ind}(\mathrm{\Sigma })\ge (2g+k-2)\mathrm{/}3$, with the sharp bound $(2g+k-1)\mathrm{/}3$ under a mild condition on the ends. The proofs use the harmonic one-form method of Ros and Chodosh–Máximo, with weighted $L^2$ spaces, adapted to the free boundary setting in the spirit of Ambrozio–Carlotto–Sharp. The main new ingredient is the computation of the dimension of the space of harmonic one-forms on a punctured compact Riemann surface with boundary which are tangential along the boundary and square integrable with respect to a weight. For a class of admissible weights $\rho$, we prove that this dimension is $2g+k-1+2{\sum }_j{N}_j-\epsilon$, where $N_j$ is the maximal order of pole allowed by $\rho$ at the $j$-th puncture and $\epsilon \in \{0,1\}$. For the weight of Chodosh–Máximo one has $N_j=d_j+1$.