https://neckpinch.com/authors/d-m-tsonev/New arXiv papers by D. M. Tsonev, from Neckpinch Tue, 10 Nov 2026 04:10:28 GMT https://neckpinch.com/papers/1706.06148/arxiv:1706.06148Mon, 19 Jun 2017 00:00:00 GMTR. R. Mesquita, D. M. Tsonev — In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain geometric operators under specified geometric flows. Given a closed, compact Riemannian manifold (Mⁿ,g(t)) and a smooth function η∈ C^(∞)(M) we consider the family of operators L=Δ− g(∇η,∇·)+cR, where R is the scalar curvature and c is some real constant. We define a geometric flow on M which encompasses the Ricci, the Ricci - Bourguignon and the Yamabe flows. Supposing that the metric g(t) evolves along this general geometric flow we derive a formula for the evolution of the eigenvalues of −L and prove monotonicity results for the eigenvalues of both −Δ+ g(∇η,∇·) and −L. We then prove Reilly-type formula for the operator L and employ it to establish an upper bound for the first variation of the eigenvalues of −L. Finally, in the pursuit of a theoretical explanation of our generalisations, we formulate two conjectures on the monotonicity of the eigenvalues of Schrödinger operators.Yamabe flowScalar curvatureEigenvalues & spectrum