It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.

We prove the equivalence between some functional inequalities and the ultracontractivity property of the heat semigroup on infinite graphs. These functional inequalities include Sobolev inequalities, Nash inequalities, Faber–Krahn inequalities, and log-Sobolev inequalities. We also show that, under the assumptions of volume growth and CDE(n, 0), which is regarded as the natural notion of curvature on graphs, these four functional inequalities and the ultracontractivity property of the heat semigroup are all true on graphs.

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In this paper, we classify n-dimensional ($n\ge 4$) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R^4 or the round cylinder S^3 x R. More generally, for $n\ge 5$, a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton $R^n$ or the product $N^{n-1}xR$, where $N^{n-1}$ is Einstein.

We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the P´olya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.