Let (M, J, Ω) be a polarized complex manifold of Kähler type. Let G be the maximal compact subgroup of the automorphism group of (M, J). On the space of Kähler metrics that are invariant under G and represent the cohomology class Ω, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the L 2-norm of the scalar curvature. We prove that the dyna...

Abstract. We consider the class of semi-stable solutions to semilinear equations −∆u = f(u) in a bounded smooth domain Ω of R (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an priori L∞ bound which holds for every semi-stable solution and every nonlinearity f . This estimate leads to the boundedness o...

whenever g > 0 is increasing and convex on R and superlinear at +∞ in the sense (7) stated below. Under these assumptions, the extremal parameter of (2) satisfies 0 < λ∗ < ∞. Here and throughout the paper, Ω is a smooth bounded domain of R . For each 0 ≤ λ < λ∗, the minimal solution of (2) is classical. Their limit as λ ↑ λ∗ is the extremal solution u∗ and it may be singular (i.e. unbounded) fo...

We argue that the paradoxes associated with infinitely degenerate states, which plague relic particle scenarios for the endpoint of black hole evaporation, may be absent when the relics are horned particles. Most of our arguments are based on simple observations about the classical geometry of extremal dilaton black holes, but at a crucial point we are forced to speculate about classical soluti...

With the question, “Can relativistic charged spheres form extremal black holes?” in mind, we investigate the properties of such spheres from a classical point of view. The investigation is carried out numerically by integrating the Oppenheimer-Volkov equation for relativistic charged fluid spheres and finding interior Reissner-Nordström solutions for these objects. We consider both constant den...

By applying a set of Hassan-Sen transformations and string dualities to the Kerr-Gödel solution of minimal D = 5 supergravity we derive a four parameter family of five dimensional solutions in type II string theory. They describe rotating, charged black holes in a rotating background. For zero background rotation, the solution is D = 5 Kerr-Newman; for zero charge it is Kerr-Gödel. In a particu...

We use geometric methods to calculate a formula for the complex Monge-Ampère measure (ddVK) n, for K Rn ⊂ Cn a convex body and VK its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same r...

We prove that the number of multigraphs with vertex set {1, . . . , n} such that every four vertices span at most nine edges is a +o(n) where a is transcendental (assuming Schanuel’s conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit ...

We construct perturbatively a class of charged black hole solutions in type 0A string theory with higher derivative terms. They have extremal limit, where the solution interpolates smoothly between near horizon AdS2 geometry and the asymptotic linear dilaton geometry. We compute the free energy and the entropy of those solution using various methods. In particular, we show that there is no corr...

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of ho...