Solving the Yamabe-Type Equations on Closed Manifolds by Iteration Schemes

TL;DR

This paper presents a novel iterative and local variational approach to solve Yamabe-type equations on closed manifolds, avoiding the use of the Weyl tensor and emphasizing the role of the conformal Laplacian's first eigenvalue.

Contribution

It introduces a double iterative scheme and local analysis method for Yamabe-type equations on closed manifolds, providing new insights without relying on the Weyl tensor.

Findings

Reproves the classical Yamabe problem via Yamabe-type equations.

Classifies solutions based on the sign of the first eigenvalue of the conformal Laplacian.

Develops a local analysis and monotone iteration scheme for solving the equations.

Abstract

We introduce a double iterative scheme and local variational method to solve the Yamabe-type equation $ - \frac{4(n - 1)}{n - 2}\Delta_{g} u + (S_{g} + \beta ) u = \lambda u^{\frac{n + 2}{n - 2}} $ for some constant $ \beta \leqslant 0 $, locally on Riemannian domain $ (\Omega, g) $ with trivial Dirichlet condition and globally on closed manifolds $ (M, g) $; the dimensions of $ \Omega $ and $ M $ are at least 3. In contrast to the traditional global variational method, these Yamabe-type equations on closed manifolds are analyzed by local analysis and monotone iteration scheme. In particular, we do not need to use the Weyl tensor. In particular, the sign of the first eigenvalue $ \eta_{1} $ of conformal Laplacian $ \Box_{g} u : = - \frac{4(n - 1)}{n - 2}\Delta_{g} u + S_{g} u $ plays the central role. When letting $ \beta \rightarrow 0 $ from the left, we reproof the classical Yamabe problem as a natural consequence of the Yamabe-type equations. The results are classified by the sign of $ \eta_{1} $.