Solving Yamabe Problem by An Iterative Method

TL;DR

This paper presents a new iterative method to solve the Yamabe problem on open domains and closed manifolds, providing a uniform proof for dimensions three and higher that goes beyond traditional minimization techniques.

Contribution

It introduces a novel iterative scheme that offers a new proof of the Yamabe problem applicable to higher dimensions, bypassing traditional functional minimization methods.

Findings

Successfully solves the Yamabe problem for n ≥ 3

Provides a uniform proof applicable to open domains and closed manifolds

Introduces an iterative method as an alternative to minimization approaches

Abstract

We introduce an iterative scheme to prove the Yamabe problem $ - a\Delta_{g} u + S u = \lambda u^{p-1} $, firstly on open domain $ (\Omega, g) $ with Dirichlet boundary conditions, and then on closed manifolds $ (M, g) $ by local argument. It is a new proof, which solves the Yamabe problem for $ n \geqslant 3 $ in a uniform argument, beyonds the traditional analysis with respect to the minimization of functionals.