Constant scalar curvature equation and the regularity of its weak solution

Abstract

In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on Kähler manifolds. We first define a notion of weak solution of CSCK for an $L^{\infty }$ Kähler metric. The main result is to show that such a weak solution (with uniform $L^{\infty }$ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of $K$-energy minimizers. The new technical ingredient is a $W^{2,2}$ regularity result for the Laplacian equation ${\Delta }_{g}u=f$ on Kähler manifolds, where the metric has only $L^{\infty }$ coefficients. It is well-known that such a $W^{2,2}$ regularity ($W^{2,p}$ regularity for any $p>1$) fails in general (except for dimension two) for uniform elliptic equations of the form $a^{ij}{\partial }_{ij}^{2}u=f$ for $a^{ij}\in L^{\infty }$, without certain smallness assumptions on the local oscillation of $a^{ij}$. We observe that the Kähler condition plays an essential role to obtain a $W^{2,2}$ regularity for elliptic equations with only $L^{\infty }$ elliptic coefficients on compact manifolds.

22 pages. Comments are welcome