The Theta Function and the Weyl Law on Manifolds Without Conjugate Points

We prove that the usual $\Theta$ function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. We deduce that the Bérard remainder in the Weyl law is valid for a manifold without conjugate points, without any restriction on the dimension.

2010 Mathematics Subject Classification: 35P20

Keywords and Phrases: Weyl law, manifolds without conjugate points, Hadamard parametrix, Jacobi and Ricatti equations

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