2022/11/28 by Jie Xu, Xu, Jie
Computer Science · Mathematics · #35J60 #53C18 #58J05 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2211.15024
openalex publication_date 2022/11/28 · openalex created_date 2022/12/10 · openalex updated_date 2026/07/28
In this article, we first show that given a smooth function S either on closed manifolds (M, g) or compact manifolds (M, g) with non-empty boundary, both for dimensions at least 3 , the condition S ≡ 0 , or S changes sign and ∫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 \citeKW2 on 2-dimensional torus, and of Escobar and Schoen \citeESS on closed manifolds for dimensions 3 and 4 . We then give results of prescribing nonzero scalar and mean curvature problems on (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.