We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ of a sequence $\mathcal{S}$ of Laplace eigenfunctions $\varphi_j$ of eigenvalue $-\lambda^2_j$ of a real analytic Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we prove that the codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ of nodal intersections with a connected, irreducible real analytic hypersurface $H \subset M$ is $\leq A_{g, H} \lambda_j$. The $\mathcal{S}$-goodness condition is that the sequence of normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly on $\mathcal{C}$, resp. $H$. We further show that a hypersurface satisfying a geometric control condition is $\mathcal{S}$-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain–Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have $L^2$ norms tending to zero.

中文翻译：

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节点相交和几何控制

我们证明结点上的$ \ mathcal数{S} $ -好实际分析的黎曼流形$（M，g）$的序列$ \ mathcal {C} $的拉普拉斯特征函数$ \ mathcal {S} $的本征值$-\ lambda ^ 2_j $的真实分析曲线$ \ mathcal {C} $由$ A_ {g，\ mathcal {C}} \ lambda_j $限制。此外，我们证明了维数为2的Hausdorff测度具有连接的，不可约实数的节点交点的$ \ mathcal {H} ^ {m-2}（\ mathcal {N} _ {\ varphi \ lambda} \ cap H）$解析超曲面$ H \ subset M $是$ \ leq A_ {g，H} \ lambda_j $。$ \ mathcal {S} $-善度条件是规格化对数序列$ \ frac {1} {\ lambda_j} \ operatorname {log} {\ lvert \ varphi_j \ rvert} ^ 2 $不会趋于$-分别在$ \ mathcal {C} $上的\ infty $。$ H $。我们进一步证明，满足几何控制条件的超曲面对于特征函数的一个子序列的密度是$ \ mathcal {S} $-良好。这部分回答了关于布尔加恩-鲁德尼克问题的超表面问题，在该表面上一系列本征函数可能消失。部分答案的特征是超曲面，在该曲面上正密度序列可以消失或仅具有趋于零的$ L ^ 2 $范数。