ON GEODESICS IN THE SPACE OF KÄHLER METRICS

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Geometric pluripotential theory on Kähler manifolds (2019, background): “20) By the examples of [83, 52] this regularity result is essentially optimal, as no higher order estimates are possible for weak solutions.”
The Monge–Ampère equation for non-integrable almost complex structures (2016, background): “[5, 24] for instances where a real Hessian bound is obtained for the complex Monge-Ampère equation, with J integrable, after having first obtained a complex Hessian bound).”
The Mabuchi Geometry of Finite Energy Classes (2014, background): “Unfortunately, smooth geodesics don’t run between the points of H [LV], however Chen proved (with complements by B locki [Bl1]) that for u0, u1 ∈ H there exists a curve [0, 1] ∋ t→ ut ∈ H∆ = PSH(X,ω) ∩ {∆u ∈ L (X)} (3) 2”
SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS (2012, unknown): “Then (1.8) max M̄ |∇2u| ≤ C1 ( 1 + max ∂M |∇2u| ) .”