The L2L^2-Alexander invariant detects the unknot

In this article, we present some of the properties of the $L^2$-Alexander invariant of a knot defined by Li and Zhang, some of which are similar to those of the classical Alexander polynomial. Notably we prove that the $L^2$-Alexander invariant detects the trivial knot.Comment: Some typos were corrected, and we added formulas for the invariant of the inverse knot or the mirror image of a kno

Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known Payne-P\'{o}lya-Weinberger and Yang universal inequalities.Comment: To appear in Calculus of variations and Partial Differential Equation

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on compact embedded strictly pseudoconvex CR manifolds

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented contact forms $\mathcal{P}_+$. We show that the functionals are continuous with respect to a natural topology on $\mathcal{P}_+$. Using a simple adaptation of the standard Kato-Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical contact forms, in a generalized sense, for the functionals. We give a necessary (also sufficient in some situations) condition for a contact form to be critical. Finally, we present explicit examples of critical contact form on both homogeneous and non-homogeneous CR manifolds.Comment: 19 pages. Comments are welcom

Water Use of Young Citrus as a Function of Irrigation Management and Ground Cover Condition

This paper describes the effect of irrigation and ground cover management on crop evapotranspiration (ET) of young Valencia citrus trees grown on Arredondo fine sand. Six different treatment combinations were used: three levels of sod water potential and two levels of ground cover conddion. Results obtained with grass cover treatments required 50% more water than with no grass cover treatments. Evapotranspiration correlated positively with the amount of irrigation applied. Month(y crop water use coefficients with grass cover treatments were 50% higher than with no grass cover treatments. Crop water use coefficient decreased as soil water potential decreased

Geometric triangulations and the Teichm\"uller TQFT volume conjecture for twist knots

We construct a new infinite family of ideal triangulations and H-triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichm\"uller TQFT and to prove the associated volume conjecture for all twist knots, using the saddle point method.Comment: v4: 90 pages, 25 figures. Comments welcome. Some of the results in this paper were announced in a note at the C. R. Acad. Sci. Paris, and some were detailed in the arXiv v1. Since v3, we made minor corrections, we added details in Section 2.9, and we added Remark 3.