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 K\"ahler manifolds. We first define a notion of weak solution of CSCK for an $L^\infty$ K\"ahler 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 K\"ahler 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 $a^{ij}\partial^2_{ij}u=f$ for $a^{ij}\in L^\infty$, without certain smallness assumptions on the local oscillation of $a^{ij}$. We observe that the K\"ahler condition plays an essential role to obtain a $W^{2, 2}$ regularity for elliptic equations with only $L^\infty$ elliptic coefficients on compact manifolds.