In this paper we construct infinitely many families of Einstein metrics on the connected sums of arbitrary number of copies of S2 × S3. We realize these 5-manifolds as total spaces of Seifert bundles over Del Pezzo orbifolds. A K¨ahler–Einstein metric on the Del Pezzo orbifold is then lifted to an Einstein metric using the Kobayashi–Boyer–Galicki method.

We consider an ancient solution g(·, t) of the Ricci flow on a compact surface that exists for t 2 (−1, T ) and becomes spherical at time t = T . We prove that the metric g(·, t) is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.

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We consider systems of partial differential equations of the first order in$t$and of order 2$s$in the$x$variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid in$R$^{n}×(0,$T$] for$s$≥2.