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Topic
CIMPA ICTP Math Seminars "Geometric Identities and their Applications on Rigidity Problems"
Date & Time
Selected Sessions:
Nov 7, 2024 03:00 PM
Description
Speaker: Dr. Allan Freitas, Universidade Federal da Paraíba, Brazil
Abstract: In recent years, Riemannian metrics that satisfy specific systemsof partial differential equations involving a function (and/or a vector field)defined in the manifold have been extensively studied. Examples includegeometric structures related to the class of static metrics that appear inGeneral Relativity, such as the critical metrics of the volume functional. Apowerful tool in this field is using integral identities (such as Reilly andPohozaev-Schoen identities), which permit relate differential and geometricoperators in Riemannian manifolds. One notable example is the proof byReilly-Ros of Alexandrov's theorem via integral inequalities, where a crucialstep is the Heintze-Karcher inequality, an inequality that follows directly fromReilly's identity and further indicates rigidity to the Euclidean ball in the caseof equality. On the other hand, the Pohozaev-Schoen identity can be used to obtain area limitations for the boundary of static manifolds and to conclude rigidity from the known models.