The Conformal Laplacian and The Kazdan-Warner Problem: Zero First Eigenvalue Case

TL;DR

This paper characterizes when scalar and Gauss curvature functions can be prescribed on manifolds with zero first eigenvalue of the conformal Laplacian, extending classical results to new geometric settings and boundary conditions.

Contribution

It provides necessary and sufficient conditions for prescribing curvature functions on manifolds with zero first eigenvalue, extending Kazdan-Warner and Escobar-Schoen results to broader cases.

Findings

Characterization of scalar curvature prescription on manifolds with zero first eigenvalue.

Extension of curvature prescription results to manifolds with boundary and zero Euler characteristic.

Generalization of the Han-Li conjecture for these geometric conditions.

Abstract

In this article, we first show that given a smooth function $ S $ either on closed manifolds $ (M, g) $ or compact manifolds $ (\bar{M}, g) $ with non-empty boundary, both for dimensions at least $ 3 $, the condition $ S \equiv 0 $, or $ S $ changes sign and $ \int_{M} S \dvol < 0 $ (with zero mean curvature if the boundary is not empty), is both the necessary and sufficient condition for prescribing scalar curvature problems within conformal class $ [g] $, provided that the first eigenvalue of the conformal Laplacian is zero. We then extend the same necessary and sufficient condition, in terms of prescribing Gauss curvature function and zero geodesic curvature, to compact Riemann surfaces with non-empty boundary, provided that the Euler characteristic is zero. These results are the first full extensions since the results of Kazdan and Warner \cite{KW2} on 2-dimensional torus, and of Escobar and Schoen \cite{ESS} on closed manifolds for dimensions $ 3 $ and $ 4 $. We then give results of prescribing nonzero scalar and mean curvature problems on $ (\bar{M}, g) $, still with zero first eigenvalue and dimensions at least $ 3 $. Analogously, results of prescribing Gauss and geodesic curvature problems on compact Riemann surfaces with boundary are given for zero Euler characteristic case. Lastly, we show a generalization of the Han-Li conjecture. Technically the key step for manifolds with dimensions at least $ 3 $ is to apply both the local variational methods, local Yamabe-type equations and a new version of the monotone iteration scheme. The key features include the smoothness of the upper solution, the technical difference between constant and non-constant prescribing scalar curvature functions, etc.