By Sebastien Boucksom, Philippe Eyssidieux, Vincent Guedj
This quantity collects lecture notes from classes provided at numerous meetings and workshops, and gives the 1st exposition in e-book kind of the elemental thought of the Kähler-Ricci move and its present state of the art. whereas numerous first-class books on Kähler-Einstein geometry can be found, there were no such works at the Kähler-Ricci move. The ebook will function a useful source for graduate scholars and researchers in advanced differential geometry, complicated algebraic geometry and Riemannian geometry, and may confidently foster additional advancements during this interesting sector of research.
The Ricci circulation used to be first brought through R. Hamilton within the early Eighties, and is valuable in G. Perelman’s celebrated evidence of the Poincaré conjecture. whilst really expert for Kähler manifolds, it turns into the Kähler-Ricci circulate, and decreases to a scalar PDE (parabolic advanced 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). presently after, G. Tian and J. music chanced on a fancy analogue of Perelman’s principles: the Kähler-Ricci stream is a metric embodiment of the minimum version application of the underlying manifold, and flips and divisorial contractions suppose the function of Perelman’s surgeries.
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Proof. Consider S D fv W Q ! t. 14). 14) in Q. 14). t; x/ D 0. s; y/ D s C p y C Xy y C ı 2 2 with ı and to be chosen later. s; y; DQ; D 2 Q/ Ä @s I/ 2 Œ 1; 1). s; y/ 2 Vr . Moreover, since . 18) to get the second inequality. 30 C. Imbert and L. s; y/ if not. 14) and U u u and U Ä uC . Consequently, U 2 S and in particular, U Ä u. tn ; xn / ! tn ; xn / ! tn ; xn / D ı > 0: This contradicts the fact that U Ä u. 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.
14). 15 below. See also Sect. 3. 14. x//j: x2Rd If non-smooth/unbounded initial data are to be considered, discontinuous stability arguments can be used next. 2 Maximal Subsolution and Bump Construction We now give more details about the general process to construct a “solution”. 0; T / for some domain Rd . 15. Assume that u˙ is a super-(resp. 14) in Q. Then there exists a function u W Q ! 14). Proof. Consider S D fv W Q ! t. 14). 14) in Q. 14). t; x/ D 0. s; y/ D s C p y C Xy y C ı 2 2 with ı and to be chosen later.
T jx s/2 2" yj2 2" Á T t : Remark that M" M > 0. This supremum is reached since u is upper semicontinuous and v is lower semi-continuous and both functions are Zd -periodic. t" ; s" ; x" ; y" / denote a maximizer. t; x/j. In particular, up to extracting p subsequences,pt" ! t, s" ! t and x" ! x, y" ! y and t" s" D O. "/ and x" y" O. "/. Assume first that t D 0. 0; x/ Ä 0: This is not possible. Hence t > 0. Since t > 0, for " small enough, t" > 0 and s" > 0. t; x/ 7! T with p" D x" y" " . t" ; x" /.