Let $M$ be a compact Riemannian manifold without boundary. We investigate the integrals of $L^2$-normalized Laplace eigenfunctions over closed submanifolds. General bounds for these quantities were obtained by Zelditch [23], and are sharp in the case where $M$ is the standard sphere. However, as with sup norms of eigenfunctions, there are many interesting settings where improvements can be made to these bounds, e.g. where $M$ is a negatively curved surface and the submanifold is a geodesic (see [6, 18]). So far, improvements in the nonpositive curvature setting have been confined to the two-dimensional case (see works of Chen and Sogge [6]; Sogge, Xi, and Zhang [18]; and the author [20, 22]). Here, we provide two theorems which extend these results into the higher dimensional setting. First, we provide an improvement of a half power of $\operatorname{log}$ over the standard bounds provided the submanifold has codimension $2$ and $M$ has strictly negative sectional curvature. Second, we provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. We use the usual tools, such as the Hadamard parametrix and the method of stationary phase, but critical to our argument is a computation of the Hessian of the distance function on the universal cover of $M$.

中文翻译：

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非正弯曲流形中的周期积分

令$ M $为无边界的紧凑黎曼流形。我们研究了闭合子流形上的$ L ^ 2 $标准化Laplace特征函数的积分。这些量的一般边界是由Zelditch [23]获得的，在$ M $是标准球体的情况下是尖锐的。但是，与本征函数的范式一样，有许多有趣的设置可以改进这些范围，例如，$ M $是负曲面，子流形是测地线（参见[6，18]）。到目前为止，非正曲率设置的改进仅限于二维情况（参见Chen和Sogge [6]； Sogge，Xi和Zhang [18]；以及作者[20，22]）。在这里，我们提供了两个定理，这些定理将这些结果扩展到更高的维度设置中。第一，如果子流形的余维数为$ 2 $，而$ M $的断面曲率严格为负，则我们提供了\\ operatorname {log} $的半数幂在标准范围内的改进。第二，对于第二基本形式与无限半径的球面有足够不同的超曲面，我们提供了相同的改进。我们使用常用的工具，例如Hadamard parametrix和固定相方法，但对我们的论点而言关键的是在$ M $的通用封面上计算距离函数的Hessian。