[Paper Review] On the constant scalar curvature Kähler metrics, existence results

Xiuxiong Chen, Jingrui Cheng|arXiv (Cornell University)|Jan 2, 2018

Geometry and complex manifolds39 references53 citations

TL;DR

The paper proves equivalence between nonexistence of cscK metrics and destabilized geodesic rays with nonincreasing K-energy, and establishes existence of cscK metrics from properness of K-energy in L1 geodesic distance, plus regularity of weak minimizers.

ABSTRACT

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth.

Motivation & Objective

• Motivate and extend a priori estimates for cscK-type equations to more general scalar curvature equations like twisted cscK.
• Prove Donaldson’s conjecture: nonexistence of cscK metric <=> destabilizing geodesic ray with nonincreasing K-energy under discrete automorphism group.
• Show that properness of K-energy in the L1 geodesic distance implies existence of a cscK metric.
• Demonstrate that weak minimizers of K-energy in the L1 space are smooth.
• Relate existence of cscK metrics to geodesic stability and to a continuous path approach for solving the cscK equation.

Proposed method

• Generalize apriori estimates from previous work to twisted cscK-type equations.
• Use Donaldson’s continuity path and its open/closedness under geometric constraints to solve the cscK equation.
• Extend K-energy and J_chi functionals to the complete geodesic metric space (E^1, d1) and study convexity along finite energy geodesics.
• Prove that K-energy properness in terms of d1 distance is equivalent to the existence of a cscK metric.
• Prove regularity: weak minimizers of (twisted) K-energy are smooth via the continuity path.

Experimental results

Research questions

• RQ1 Does the nonexistence of a cscK metric imply the existence of a destabilizing geodesic ray with nonincreasing K-energy?
• RQ2 Does properness of the K-energy with respect to the L1 geodesic distance guarantee the existence of a cscK metric?
• RQ3 Are weak minimizers of the (twisted) K-energy in (E^1, d1) necessarily smooth?
• RQ4 How does twisted K-energy interact with geodesic stability and the Donaldson conjecture in the discrete automorphism setting?
• RQ5 Can the continuity path be used to extend existence results to twisted cscK equations and to general automorphism groups?

Key findings

• Under the assumption Aut0(M,J)=0, Donaldson’s conjecture is established: nonexistence of a cscK metric is equivalent to the existence of a destabilized geodesic ray with nonincreasing K-energy.
• The existence of a cscK metric is equivalent to the properness of the K-energy with respect to the L1 geodesic distance d1 on the space of Kähler potentials.
• Weak minimizers of the K-energy in (E^1, d1) are smooth; this regularity result holds for twisted K-energy as well.
• The K-energy and J_chi can be extended to (E^1, d1) with convexity along finite energy geodesics, enabling variational approaches to cscK existence.
• A compactness theorem shows that a set of Kähler potentials with bounded scalar curvature and entropy is precompact in C^{3,α}, leading to regularity results and Calabi flow extension under curvature bounds.
• The work introduces and uses the invariant R([ω0], [χ]) to quantify solvability of the twisted continuity path and shows R is well-defined and invariant in the same Kähler class.
• If the K-energy is bounded below, then R([ω0],[χ])=1 if and only if R([ω0],[χ])>0, clarifying when the twisted path can be solved up to t<1.