By Vincent Guedj, Ahmed Zeriahi
Advanced Monge–Ampère equations were the most strong instruments in Kähler geometry due to the fact Aubin and Yau’s classical works, culminating in Yau’s strategy to the Calabi conjecture. A awesome program is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. lately degenerate advanced Monge–Ampère equations were intensively studied, requiring extra complex tools.
The major target of this booklet is to offer a self-contained presentation of the new advancements of pluripotential concept on compact Kähler manifolds and its software to Kähler–Einstein metrics on mildly singular types. After reviewing simple houses of plurisubharmonic features, Bedford–Taylor’s neighborhood conception of advanced Monge–Ampère measures is constructed. with the intention to clear up degenerate complicated Monge–Ampère equations on compact Kähler manifolds, fantastic houses of quasi-plurisubharmonic capabilities are explored, sessions of finite energies outlined and diverse greatest ideas demonstrated. After proving Yau’s celebrated theorem in addition to its fresh generalizations, the implications are then used to unravel the (singular) Calabi conjecture and to build (singular) Kähler–Einstein metrics on a few types with gentle singularities.
The publication is available to complicated scholars and researchers of advanced research and differential geometry.
Keywords: Pluripotential thought, plurisubharmonic capabilities, advanced Monge–Ampère operators, generalized capacities, susceptible suggestions, a priori estimates, canonical Kähler metrics, singular kinds
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Additional resources for Degenerate Complex Monge–Ampère Equations
A C seiÂ / D 0 for all Â 2 Œ0; 2 , hence h is constant, as claimed. 2. a/ D maxDN h. If a 2 D, the previous case shows that h is constant in a neighborhood of a. a/ D maxDN hg is open, non-empty and closed (by continuity). We infer A D D, hence h is constant in D. b C be a bounded domain and W @ ! R a continuous function (the boundary data). The Dirichlet problem for the homogeneous Laplace equation consists in finding a harmonic function h W ! R that is a solution of the following linear PDE with prescribed boundary values: h D 0 in ; DirMA.
A C r / d . a C s / d . a C r / d n . 4) Tn where d n is the normalized Lebesgue measure on the torus Tn . All these integrals make sense in Œ 1; C1Œ. 34). Proof. 1) by integration over the unit sphere in Cn and the second inequality follows from the first one by integration over Œ0; r against the measure d . 1) by integration on the torus. 32. Let u be a plurisubharmonic in . a C r /d . a C t /d . a C r / d V . 5) j j<1 where Ä2n denotes the volume of the unit ball in Cn . The function u, considered as a function on 2n real variables, is thus subharmonic in considered as a domain in R2n .
3. Poisson formula/. Let h W D ! R be a continuous function which is harmonic in D. eiÂ / 0 1 jzj2 dÂ: jeiÂ zj2 6 1 Plurisubharmonic functions Proof. We reduce to the case when h is harmonic in a neighborhood of D by considering z 7! rz/ for 0 < r < 1 and letting r increase to 1 in the end. For z D 0 the formula above is the mean value property in the unit disc. z/ yields z D f a . a/ D 2 j1 ae N iÂ j2 0 The change of variables as desired. 4. Let h W D ! RC be a non-negative continuous function which is harmonic in D.
Degenerate Complex Monge–Ampère Equations by Vincent Guedj, Ahmed Zeriahi