完备非紧自由边界极小曲面的指标估计

中文摘要 AI 辅助

设 $\Omega\subset\mathbb R^3$ 为具有光滑边界 $\partial\Omega$ 的无界区域，$\Sigma$ 为 $\Omega$ 中完备、非紧、可定向的浸入自由边界极小曲面，其边界 $\partial\Sigma\subset\partial\Omega$ 紧致且具有有限 Morse 指标。我们证明，若 $\partial\Omega$ 的平均曲率沿 $\partial\Sigma$ 满足 $H_{\partial\Omega}\ge 0$，且在 $\partial\Sigma$ 的某点处 $H_{\partial\Omega}>0$，则 \\[ \mathrm{Ind}(\Sigma)\\ \ge\\ \frac13\Bigl(2g+k+2\sum_{j=1}^r(d_j+1)-2\Bigr), \\] 其中 $g$ 为 $\Sigma$ 的亏格，$k$ 为 $\partial\Sigma$ 的连通分支数，$r$ 为 $\Sigma$ 的端数，$d_1,\dots,d_r$ 分别为各端的重数。当仅假设 $H_{\partial\Omega}$ 非负时，我们得到 $\mathrm{Ind}(\Sigma)\ge(2g+k-2)/3$，并在关于端的温和条件下得到锐界 $(2g+k-1)/3$。证明使用 Ros 与 Chodosh--Máximo 的调和一次形式方法，结合加权 $L^2$ 空间，并按照 Ambrozio--Carlotto--Sharp 的精神适应自由边界情形。主要新成分是计算带边界的穿孔紧黎曼曲面上沿边界切向且关于权重平方可积的调和一次形式空间的维数。对于一类可容许权重 $\rho$，我们证明该维数为 $2g+k-1+2\sum_jN_j-\varepsilon$，其中 $N_j$ 为 $\rho$ 在第 $j$ 个穿孔处允许的极点最大阶数，$\varepsilon\in\{0,1\}$。对于 Chodosh--Máximo 的权重，有 $N_j=d_j+1$。

英文摘要

Let $Ω\subset\mathbb R^3$ be an unbounded domain with smooth boundary and let $Σ$ be a complete, noncompact, orientable, immersed free boundary minimal surface in $Ω$, with compact boundary $\partial Σ\subset\partial Ω$ and finite Morse index. We prove that if the mean curvature of $\partial Ω$ satisfies $H_{\partial Ω}\ge 0$ along $\partial Σ$ and $H_{\partialΩ}>0$ at some point of $\partialΣ$, 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 $Σ$, $k$ is the number of connected components of $\partialΣ$, $r$ is the number of ends of $Σ$ and $d_1,\dots,d_r$ are their respective multiplicities. When $H_{\partialΩ}$ is only assumed to be nonnegative we obtain $\textrm{Ind}(Σ)\ge(2g+k-2)/3$, 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 $ρ$, we prove that this dimension is $2g+k-1+2\sum_jN_j-\varepsilon$, where $N_j$ is the maximal order of pole allowed by $ρ$ at the $j$-th puncture and $\varepsilon\in\{0,1\}$. For the weight of Chodosh--Máximo one has $N_j=d_j+1$.

发表机构

• Institute of Mathematics, Federal University of Alagoas (UFAL)（阿拉戈斯联邦大学数学研究所）

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