By Sebastien Boucksom, Philippe Eyssidieux, Vincent Guedj

This quantity collects lecture notes from classes provided at numerous meetings and workshops, and offers the 1st exposition in e-book type of the elemental thought of the Kähler-Ricci circulate and its present cutting-edge. whereas numerous very good books on Kähler-Einstein geometry can be found, there were no such works at the Kähler-Ricci movement. The publication will function a worthwhile source for graduate scholars and researchers in complicated differential geometry, complicated algebraic geometry and Riemannian geometry, and should expectantly foster additional advancements during this interesting zone of research.

The Ricci movement was once first brought by means of R. Hamilton within the early Eighties, and is relevant in G. Perelman’s celebrated evidence of the Poincaré conjecture. whilst really good for Kähler manifolds, it turns into the Kähler-Ricci stream, and decreases to a scalar PDE (parabolic complicated Monge-Ampère equation).

As a spin-off of his step forward, G. Perelman proved the convergence of the Kähler-Ricci circulation on Kähler-Einstein manifolds of confident scalar curvature (Fano manifolds). almost immediately after, G. Tian and J. music found a fancy analogue of Perelman’s principles: the Kähler-Ricci circulate is a metric embodiment of the minimum version software of the underlying manifold, and flips and divisorial contractions think the function of Perelman’s surgeries.

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**Example text**

The proof of the lemma is now complete. 3 Continuous Solutions from Comparison Principle As mentioned above, the maximal subsolution u is not necessarily continuous; hence, its lower semi-continuous envelope u does not coincide necessarily with it. 4 (cf. 13 above). We would get a (continuous viscosity) solution if u D u . On one hand, u is upper semi-continuous by construction and on the other hand u Ä u by definition of the semi-continuous envelopes. 14) if and only if u Ä u in Q. 14) satisfies a comparison principle and that the barriers u˙ satisfy some appropriate inequality on the parabolic boundary.

The fundamental assumptions is that the equation is uniformly elliptic (see below for a definition). For pedagogical purposes, we do not want to prove a theorem for the most general case. x/jDuj @t trace. 25) Assumptions (A) • c is bounded and Lipschitz continuous in Q; • W Q ! e. 29. 25) in an open set Q Rd C1 is H¨older continuous in time and space. 2"/ 1 jx yj2 . Such a result can be found in [CIL92]. 30 (Jensen–Ishii’s Lemma II). Let U and V be two open sets of Rd and I an open interval of R.

4 (cf. 13 above). We would get a (continuous viscosity) solution if u D u . On one hand, u is upper semi-continuous by construction and on the other hand u Ä u by definition of the semi-continuous envelopes. 14) if and only if u Ä u in Q. 14) satisfies a comparison principle and that the barriers u˙ satisfy some appropriate inequality on the parabolic boundary. More precisely, we would like on one hand that Comparison principle. 14) in Q and u Ä v on the parabolic boundary @p Q, then u Ä v in Q.