Microlocal decoupling inequalities and the distance problem on Riemannian manifolds. (English) Zbl 1511.53045
In geometric measure theory, Falconer’s conjecture is an unsolved problem concerning the sets of Euclidean distances between points in compact \(d\)-dimensional spaces, see [K. J. Falconer, Mathematika 32, 206–212 (1985; Zbl 0605.28005)]. The Falconer distance problem played an important role in several classical problems of geometric measure theory and harmonic analysis, see, e.g., [P. Shmerkin, Isr. J. Math. 230, No. 2, 949–972 (2019; Zbl 1414.28011); T. Orponen, Adv. Math. 307, 1029–1045 (2017; Zbl 1355.28018); Nonlinearity. 25, No. 6, 1919–1929 (2012; Zbl 1244.28014); P. Mattila and T. Orponen, Proc. Am. Math. Soc. 144, No. 8, 3419–3430 (2016; Zbl 1345.28008); L. Guth et al., Invent. Math. 219, No. 3, 779–830 (2020; Zbl 1430.28001); A. Iosevich and S. Senger, Ann. Acad. Sci. Fenn. Math. 41, No. 2, 713–720 (2016; Zbl 1345.28006)].
The main goal of this paper is to study a generalization of the Falconer distance problem to the Riemannian setting. The authors extend the result of L. Guth et al. [Invent. Math. 219, No. 3, 779–830 (2020; Zbl 1430.28001)] for the distance set in the plane to general Riemannian surfaces.
The main goal of this paper is to study a generalization of the Falconer distance problem to the Riemannian setting. The authors extend the result of L. Guth et al. [Invent. Math. 219, No. 3, 779–830 (2020; Zbl 1430.28001)] for the distance set in the plane to general Riemannian surfaces.
Reviewer: Lakehal Belarbi (Mostaganem)
MSC:
| 53C22 | Geodesics in global differential geometry |
| 28A75 | Length, area, volume, other geometric measure theory |
Citations:
Zbl 0605.28005; Zbl 1414.28011; Zbl 1355.28018; Zbl 1244.28014; Zbl 1345.28008; Zbl 1430.28001; Zbl 1345.28006References:
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