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Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jfa.2020.108866
Victor Kalvin

Abstract We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area 4π the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.

中文翻译:

锥奇点黎曼曲面上拉普拉斯算子行列式的 Polyakov-Alvarez 型比较公式

摘要 我们提出并证明了对应于具有圆锥奇点的紧黎曼曲面上的共形等效度量的拉普拉斯算子的 Friederichs 扩展的行列式的 Polyakov-Alvarez 型比较公式。特别是,我们发现行列式如何取决于圆锥奇点的阶数。我们还通过几个例子来说明这些一般结果:基于我们的 Polyakov-Alvarez 类型公式,我们恢复已知并获得新的明确公式,用于在有边界和无边界的奇异表面上拉普拉斯算子的行列式。在其中一个例子中,我们表明,在具有两个圆锥奇点和固定面积 4π 的球面上的恒定曲率度量上,弗里德里希拉普拉斯算子的行列式从上方是无界的,并在标准圆形球的度量上达到其局部最大值。
更新日期:2021-04-01
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