Constant Scalar Curvature Equation and Regularity of Its Weak Solution

Abstract

In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth‐order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ 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 W2, 2 regularity result for the Laplacian equation Δgu=f on Kähler manifolds, where the metric has only L∞ coefficients. It is well‐known that such a W2, 2 regularity (W2, p regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form aij∂ij2u=f for aij ∊ L∞ without certain smallness assumptions on the local oscillation of aij. We observe that the Kähler condition plays an essential role in obtaining a W2, 2 regularity for elliptic equations with only L∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.

Approximation of weak geodesics and subharmonicity of Mabuchi energy, II: $ε$-geodesics

“…Suppose the two boundaries ϕ 0 , ϕ 1 of G are both nondegenerate energy minimizers of M. Then M(ϕ t ) keeps to be a constant along this geodesic, and it satisfies the ε-affine condition automatically. Therefore, our Theorem (5.4) implies that the geodesic G is fiberwise uniformly non-degenerate, and then the regularities of G can be improved by the work of He-Zeng( [19]). Then we recover one of our result in [20].…”

“…If we allow that the boundary of G has merely C 1, 1-regularities, then the answer is negative due to the example of Berman ([4]). On the other hand, the answer is affirmative ( [20]), when G is connecting with two nondegenerate energy minimizers of M. In fact, the boundary of G must be smooth in the later case by the work of He-Zeng ( [19]). Therefore, a geodesic G will always be assumed to have C 1, 1-regularities and its boundary belongs to the space H through out this paper.…”

Strict convexity of the Mabuchi functional for energy minimizers

“…Moreover, we prove that such a energy minimizer satisfies the weak cscK equation in the sense of He-Zeng ( [13]), and He-Zeng's regularities estimates enable us to improve the regularities of the geodesic on the fiber direction. In fact, the restriction G| Xt must be a smooth cscK metric for each t ∈ [0, 1].…”