Nicholas J. Korevaar

Refined asymptotics for constant scalar curvature metrics with isolated singularities

Abstract. We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, , in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, [2], we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohožaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions.

CONSTANT MEAN-CURVATURE SURFACES IN HYPERBOLIC SPACE

Supported by the National Science Foundation grant DMS-8808002. Supported by the National Science Foundation grant DMS-8908064. The research described in this paper was supported by research grant DE-FG0286ER250125 of the Applied Mathematical Science subprogram of Office of Energy Research, U.S. Department of Energy, and National Science Foundation grant DMS-8900285. Supported by the National Science Foundation grant DMS-8800414.

A priori interior gradient bounds for solutions to elliptic Weingarten equations

Abstract In this paper a maximum principle approach is used to derive a priori interior gradient bounds for smooth solutions to the Weingarten equations f ( λ ) = ∑ i 1 i 2 … i k λ i 1 … λ i k = ψ ( x , u , v ) . Here λ = (λ 1 , …, λ n ) is the vector of principal curvatures of the graph of u at a point ( x , u ( x )) on the graph, with downward normal v . One requires a one-sided height bound ( u f is elliptic. The result generalizes what is known to be true for the prescribed mean curvature equation ( k = 1).

Maximum principle gradient estimates for the capillary problem

On utilise le principe de maximum pour estimer le gradient des solutions du probleme capillaire non parametrique a n dimensions. On demontre que des solutions sont continues de Lipschitz dans certains domaines singuliers