By Vincent Guedj, Ahmed Zeriahi
Complicated Monge–Ampère equations were some of the most strong instruments in Kähler geometry considering that Aubin and Yau’s classical works, culminating in Yau’s option to the Calabi conjecture. A outstanding software is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. in recent times degenerate advanced Monge–Ampère equations were intensively studied, requiring extra complex tools.
The major aim of this ebook is to provide a self-contained presentation of the hot advancements of pluripotential thought on compact Kähler manifolds and its software to Kähler–Einstein metrics on mildly singular forms. After reviewing easy houses of plurisubharmonic services, Bedford–Taylor’s neighborhood concept of complicated Monge–Ampère measures is built. on the way to remedy degenerate complicated Monge–Ampère equations on compact Kähler manifolds, effective homes of quasi-plurisubharmonic capabilities are explored, periods of finite energies outlined and diverse greatest ideas validated. 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 forms with light singularities.
The booklet is on the market to complex scholars and researchers of complicated research and differential geometry.
Keywords: Pluripotential conception, plurisubharmonic features, complicated Monge–Ampère operators, generalized capacities, vulnerable suggestions, a priori estimates, canonical Kähler metrics, singular forms
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Additional info for Degenerate Complex Monge–Ampère Equations
Let f W ! C be a holomorphic function such that f 6Á 0. Then log jf j 2 SH. /. 0/ is the zero set of f in of f at the point a. a/ is the order of vanishing Observe that since f 6Á 0, the zero set Zf is discrete in locally finite. , hence the sum is Proof. z/ D a2A for z 2 D. a/ıa d d c log jf j D a2A in the sense of distributions. a2A 16 1 Plurisubharmonic functions be a Borel measure with compact support on C, then Z log jz jd . / D ? z/ z 7 ! z/ WD Let C is subharmonic in C. Moreover, if z 2 C n Supp.
We will provide more precise information on their size in the next chapters. Proof. Fix u 2 PSH. / et let G denote the set of points a 2 such that u is integrable in a neighborhood of a. We are going to show that G is a non-empty open and closed subset of . It will follow that G D (by connectedness) and u 2 L1loc . /. Note that G is open by definition. a; r/. a/ > 1, then a 2 G, hence G ¤ ;, since u 6Á 1. 22 1 Plurisubharmonic functions We finally prove that G is closed. b; r/ b . b; r/. a0 / > 1.
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.
Degenerate Complex Monge–Ampère Equations by Vincent Guedj, Ahmed Zeriahi