“QUINZENA DE GEOMETRIA” Dragomir Tsonev (UFAM, Manaus) Title: ON THE SPECTRA OF GEOMETRIC OPERATORS EVOLVING WITH GEOMETRIC FLOWS

Dragomir Tsonev (UFAM, Manaus) Title: ON THE SPECTRA OF GEOMETRIC OPERATORS EVOLVING WITH GEOMETRIC FLOWS Abstract: 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 compact Riemannian manifold (M, g(t)) and a smooth function η ∈ C(M) we consider the family of operators∆−g(∇η,∇·)+cR whereR is the scalar curvature and c is some real constant. We define a geometric flow onM which encompasses the Ricci, the Ricci Bourguignon and the Yamabe flows. Supposing that themetric g(t) evolves along this general geometric flowwe first derive a formula for the evolution of the eigenvalues λ(t). Moreover, under certain restrictions on the aforementioned flow as well some curvature assumptions, we prove a monotonicity results for the eigenvalues of both the Witten Laplacian ∆− g(∇η,∇·) and the more general operator∆− g(∇η,∇·)+ cR. We also derive Bocher and Reilly type formulas for the latter operator and, assuming that the manifold M is closed, an estimate for λ(t) is given.