3.22 Tesis doctorado

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Browsing 3.22 Tesis doctorado by Author "Espinal Florez, María Fernanda"

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    Problems on conformal invariance and Yamabe-Type flows
    (2024) Espinal Florez, María Fernanda; Sáez Trumper, Mariel; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    This work is specifically focused on the study of quantities in Riemannian geometry under a conformal change of metric, that is, under changes of metric which stretch the length of vectors but preserve the angle between any pair of vectors. In this context, my thesis work has centered on the study of symmetric polynomials σk of the eigenvalues of the Schouten tensor, which satisfy a tranformation law under conformal changes. This work consists of two parts. The rst problem concentrates on Yamabe-type ows for σk-curvature, which are classic examples of intrinsic non-linear geometric ows. Inspired by work of Daskalopoulos and Sesum [22], we investigate the existence and classi cation of conformally at rotationally symmetric k-Yamabe gradient solitons replacing scalar curvature by σk-curvature. Our rst result reduces the classi cation of k-Yamabe solitons to the classi cation of global smooth solutions of a fully nonlinear elliptic equation. Regarding the existence result, through a phase-plane analysis of an autonomous system of ordinary equations as in [71], we were able to prove local existence of the ow under conditions of admissibility for the initial metric when n ≥ 2k. Additionally, we had to analyze the asymptotic behavior and solution pro le in each case, taking into account, especially the admissibility of the solution. In contrast with the classical case, the fully non-linear nature of the problem requires additional restrictions (to ensure admissibility) and a more delicate analysis. On the other hand, in collaboration with Professor M. González [27] we work on the k-Yamabe singular problem. The research was focused on constructing metrics with constant σ2-curvature and non-isolated singularities. Speci cally, we contructed a complete non-compact Riemannian metrics with positive constant σ2-curvature on the sphere Sn with a prescribed singular set Λ given by a disjoint union of closed submanifolds whose dimension is positive and strictly less than n−√n−2 2 . This is a fully non-linear problem, nevertheless, we show that the classical gluing method (used by Mazzeo-Pacard for the scalar curvature [56]) still works in this setting since the linearized operator has good mapping properties in weighted spaces. The idea to construct this metric is to nd rst an approximate metric with the right asymptotic behavior near the singularity. Even though many of our arguments would work for a general k, we have some computational di culties that restrict our theorem to k = 2.