By Lucien Guillou, Alexis Marin

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Extra info for A la recherche de la topologie perdue: I Du côté de chez Rohlin, II Le côté de Casson

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Lemma 2. Let u < 0 on 8B1, det D2u > 1. Then inf6, u < -1/2. Proof. u < (IxI2 - 1). Here it is easy to see that the right hand side is a barrier 2 and the inequality follows as in the proof of Lemma 1. Corollary 1. If a) 0 < Al < Mu < 1\2 in fZ, with B1 C tl c BK. b) u=0on8C. c) infu >b>0. Then distance (Xo, 8fl) > µ(b, K, A,, A2) > 0. Theorem 1. Assume u > 0 satisfies (1) and that the convex set {u = 0} is not a point. Then {u = 0) cannot have extremal points in the interior of the domain of definition of u.

PART 3. A PRIORI ESTIMATES OF SOLUTIONS TO MONGE AMPERE EQUATIONS 43 Figure 2 (Xo, U(Xo)) "any vector v with Iv'I < Cu(Xo) and u(Xo) 0, along BitZ) .

E. e. I v (u - 4E) 1 > 0, along BitZ) . We then get the same contradiction to Lemma 1 from the renormalization of StE(uk) for k large enough. We next show how a careful normalization of the argument in proving strict convexity implies Cl,a regularity. The main lemma is the following. Lemma 2. Let u be a solution of detD,,u=dp on fl normalized as follows a) u=1 onOf,BICftCB b) infn u = u(Xo) = 0 c) µ satisfies property P1 and hence a(ft) N 1 (from Lemma 1). e. a) h,(X - X0) is homogeneous of degree one and b) h0(X-Xo)=aforXE{u=a}.

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