Chapter 14

Monge–Ampère equations and stabilityABAD0B75D26A

Smooth, continuous, degenerate, and prescribed-singularity equations.

Sections and subsections

1. 14.1 Smooth Monge–Ampère equations92B1F482D72A
2. 14.2 Scalar bounds and uniqueness5E342E12EF34
3. 14.3 The potential bound from weighted integrationD57A1C84047D
4. 14.4 The linearized equation and smooth dependence7BDF4C42EB78
5. 14.5 Second derivatives and ellipticityCBFCDA7183ED
6. 14.6 The continuity methodF1D5D01E2624
7. 14.7 The smooth exponential equationA4D09B787D84
8. 14.8 Variation of the normalization scalar9541740AAFDC
9. 14.9 Normalization on the Hopf threefold44E03525BBAF
10. 14.10 Prescribing a Gauduchon volume form6950E7B1FBCB
11. 14.11 Bounds for the Gauduchon potentialF9FF7B5E741A
12. 14.12 The adjoint equation for Gauduchon volumeF6A1B32989A1
13. 14.13 The matrix structure of the Gauduchon equation53E844FE515C
14. 14.14 The largest-eigenvalue estimate148E6041EC3F
    1. 14.14.2 Differentiating the largest eigenvalue.9994CF39B2B0
    2. 14.14.6 The third derivatives and the torsion term.D5E11D27FAAA
    3. 14.14.12 Cancellation when the smallest eigenvalue is controlled.DA6F05F8F203
    4. 14.14.15 The remaining eigenvalue configuration.57C7A82680F3
15. 14.15 A Liouville theorem for the limiting coneD4DE203BE3BF
16. 14.16 The gradient bound by rescaling40669ABC11F8
17. 14.17 Existence for the Gauduchon ansatz6904C64D5831
18. 14.18 Normalization for the Gauduchon equation9C1043361EBF
19. 14.19 Continuous solutions and normalization080A6052D542
    1. 14.19.1 Continuous solvability49F0FF1AED46
    2. 14.19.5 The modified comparison principle88B9A2247D18
    3. 14.19.13 Capacity estimates and uniform compactnessEF206AD1915F
    4. 14.19.22 Stability under a power capacity bound042B61EBF79D
    5. 14.19.25 Construction of continuous solutionsFEE6FC2B120F
    6. 14.19.30 Smoothness of bounded solutions6CE6BC329E93
    7. 14.19.33 The vanishing exponent and scalar uniqueness27FAAF143258
    8. 14.19.37 Normalization on the Hopf threefold4D6684B2CEDB
    9. 14.19.42 Stability for concave equationsA73917A5BC62
    10. 14.19.47 Gradient bounds for concave equationsF77131C8889A
20. 14.20 Uniqueness for degenerate equations6B1310DDF7B5
21. 14.21 Extrema and density sublevels6267D9D97C43
    1. 14.21.12 A square-root threshold95F071B4092F
22. 14.22 Polarized cofactors and propagation9769545E981A
    1. 14.22.3 Weighted energy1961884C0812
    2. 14.22.8 The real symbol and exterior-ball propagation1FF2DD5C252C
    3. 14.22.12 The unweighted energy defectB7138A0B3C69
23. 14.23 Integral rigidity in dimensions two and three4DE5082C1B2B
24. 14.24 Bounded uniqueness and connectednessE0DB623401C5
    1. 14.24.4 The energy of the difference42826D458068
    2. 14.24.12 Concentrating smooth test potentialsE597C8D6FBF4
25. 14.25 Comparison and invariant mass48CE016DFF65
26. 14.26 Smooth nonuniqueness for a degenerate densityD85397FA2C2A
    1. 14.26.2 Three rank-one directions0CFD28034339
    2. 14.26.7 The scalar coefficients68DF63E92045
    3. 14.26.9 Two forms with the same cubeDBEDC79E97CD
    4. 14.26.10 Restoring strict positivity of the background2D54B6D6C7AD
    5. 14.26.12 A continuum of radial solutions6FDC265145E3
27. 14.27 Nonuniqueness with a zero set of empty interior5278ECB27C50
28. 14.28 Quadratic density zeros9ABE80D68F2B
29. 14.29 A real analytic densityC3B76FE21E71
    1. 14.29.4 Prescribing the determinantCB8BE6BADB1A
    2. 14.29.11 Extension over the projective endsEA62A44D2D9F
    3. 14.29.12 A positive equation background38D1973448C7
    4. 14.29.14 Zeros and higher dimensionsE82948C4DB2A