By Vincent Guedj, Ahmed Zeriahi
Advanced Monge–Ampère equations were the most robust instruments in Kähler geometry given that Aubin and Yau’s classical works, culminating in Yau’s strategy to the Calabi conjecture. A impressive software is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. in recent times degenerate complicated Monge–Ampère equations were intensively studied, requiring extra complicated tools.
The major aim of this booklet is to offer a self-contained presentation of the new advancements of pluripotential thought on compact Kähler manifolds and its software to Kähler–Einstein metrics on mildly singular kinds. After reviewing easy homes of plurisubharmonic capabilities, Bedford–Taylor’s neighborhood concept of advanced Monge–Ampère measures is constructed. that allows you to remedy degenerate complicated Monge–Ampère equations on compact Kähler manifolds, tremendous homes of quasi-plurisubharmonic capabilities are explored, periods of finite energies outlined and numerous greatest rules proven. After proving Yau’s celebrated theorem in addition to its fresh generalizations, the implications are then used to resolve the (singular) Calabi conjecture and to build (singular) Kähler–Einstein metrics on a few forms with light singularities.
The booklet is out there to complicated scholars and researchers of complicated research and differential geometry.
Keywords: Pluripotential concept, plurisubharmonic services, advanced Monge–Ampère operators, generalized capacities, vulnerable options, a priori estimates, canonical Kähler metrics, singular forms
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Extra resources for Degenerate Complex Monge–Ampère Equations
0/ D 2 j j<1 0 where d . z/ D u. / @D WD u is the 2 1 jzj2 d . /: jz j2 This is a negative harmonic function in the unit disc. 0/ for jzj < 1=2. z/ p Ä 3. 4 Hartogs lemma and the Montel property 31 jz aj . This is a positive harmonic function in D n fag j1 z aj N with a logarithmic singularity at a. 4=3//p : Cp Assume now jaj > 3=4. 3=4/. 1=jaj/; for jzj Ä 1=2. 1=j j/d . 3) that the set of points where jujp is locally integrable is a non-empty, open, and closed in , hence u 2 Lploc . /. Recall now that uj !
A/ D maxDN u. z/ N D max ug N D is open, non-empty and closed by upper semi-continuity, hence u is constant in D. 16. Let b C a bounded domain and u a subharmonic function in Assume that lim supz! z/ Ä 0 for all 2 @ . Then u Ä 0 in . Proof. Fix " > 0. By the compactness and upper semi-continuity of u, there exists a compact subset K such that u Ä " in n K. 15 to conclude that u Ä " in D. Therefore, u Ä " in and the conclusion follows since " > 0 is arbitrary. 17. Let b C be a bounded domain and u; v subharmonic functions in L1loc .
Compute the Laplacian in polar coordinates in C. 2. jzj/, a smooth function in Œ0; RŒ. r/: r 3. Describe all harmonic radial functions in C. 4. 0; R/ iff function of t D log r in the interval 1; log RŒ. 7. Let h W R2 ! R be a harmonic function. x/j Ä C Œ1 C jjxjjd ; for all x 2 R2 : Show that h is a polynomial of degree at most d . 8. Let W @ ! R be a continuous function and let u denote the Poisson envelope of in the unit disc C. i) Show that u is H¨older continuous on if and only if is H¨older continuous.