Index estimates for complete noncompact free boundary minimal surfaces

Abstract

Let $\Omega \subset {\mathbb{R}}^3$ be an unbounded domain with smooth boundary and let $\Sigma$ be a complete, noncompact, orientable, immersed free boundary minimal surface in $\Omega$, with compact boundary $\partial \Sigma \subset \partial \Omega$ and finite Morse index. We prove that if the mean curvature of $\partial \Omega$ satisfies $H_{\partial \Omega }\ge 0$ along $\partial \Sigma$ and $H_{\partial \Omega }>0$ at some point of $\partial \Sigma$, then \[ \textrm{Ind}(Σ)\ \ge\ \frac13\Bigl(2g+k+2\sum_{j=1}^r(d_j+1)-2\Bigr), \] where $g$ is the genus of $\Sigma$, $k$ is the number of connected components of $\partial \Sigma$, $r$ is the number of ends of $\Sigma$ and $d_1,\dots ,d_r$ are their respective multiplicities. When $H_{\partial \Omega }$ is only assumed to be nonnegative we obtain $\text{Ind}(\Sigma )\ge (2g+k-2)/3$, with the sharp bound $(2g+k-1)/3$ 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 $N_j=d_j+1$.

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