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There are many examples of several variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger [D] and, later, by Rodriguez-Villegas [R] and Maillot [M]. While Rodriguez-Villegas made this relationship explicit for the two-variable case, it is our goal to understand the three-variable case and shed some light on the examples with more variables
The hyperbolic metric for the punctured unit disc in the Euclidean plane is singular at the origin. A renormalization of the metric at the origin is provided by the Euclidean metric. For Riemann surfaces, there is a unique germ for the isometry class of a complete hyperbolic metric at a cusp. The renormalization of the metric for the punctured unit disc provides a renormalization for a hyperbolic metric at a cusp. For a holomorphic family of punctured Riemann surfaces, the family of (co)tangent spaces along a puncture defines a tautological holomorphic line bundle over the base of the family. The Hermitian connection and Chern form for the renormalized metric are determined. Connections to the works of M. Mirzakhani [Mi1], [Mi2] and L. Takhtajan and P. Zograf [TZ2] and to intersection numbers for the moduli space of punctured Riemann surfaces studied by E. Witten [Wi1], [Wi2] are presented
We estimate the -norm () of the restriction to a curve of the eigenfunctions of the Laplace-Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere, these estimates are sharp. If the curve has nonvanishing geodesic curvature, we can improve our results. All our estimates are shown to be optimal for the sphere. Moreover, we sketch their extension to higher dimensions.
On prouve une estimation de la norme () de la restriction à une courbe des fonctions propres de l'opérateur de Laplace-Beltrami sur une surface riemannienne. Si la courbe est une géodésique de la sphère, on montre que nos estimations sont optimales. En revanche, si la courbe possède une courbure géodésique non nulle, on améliore le résultat. Toutes nos estimées sont optimales sur la sphère. Nous en esquissons par ailleurs des généralisations aux dimensions supérieures
We consider the critical Korteweg–de Vries (KdV) equation: Let () be soliton solutions to this equation. Denote the KdV linear group, and let be with sufficient decay on the right; that is, let be for some .
We construct a solution to the critical KdV equation such that
Let be a reductive algebraic group defined over a number field , and let be a finite set of valuations of containing all archimedean ones. Let , and let be an -arithmetic subgroup of . Let and , where each is a torus of of maximal -rank. We prove that if admits a closed -orbit, then either or is a singleton, and we describe the closed -orbits in both cases. We apply this result to prove that if a collection of decomposable homogeneous forms takes discrete values at , where is the ring of -integers of , then there exists a homogeneous form such that , , for all . Our result is also new in the simplest case of one real homogeneous form when and
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