Surfaces with radially symmetric prescribed Gauss curvature · arXivDesk
math/9906114Jun 16, 1999revised version; 44 pages; 2 figures; The published version of this preprint (see the journal reference field) appeared under the title: Surfaces with prescribed Gauss curvature
Surfaces with radially symmetric prescribed Gauss curvature
We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions u of the PDE: Δu(x)+K(x)exp(2u(x))=0, with K(x)
Nearby in the stack
the Gauss curvature function at
x∈²
. We assume that the integral curvature is finite. For radially symmetric
K
we introduce the notion of a least integrally curved surface, and also the notion of when such a surface is critical. With respect to these notions we analyze the radial symmetry of
u
for the whole spectrum of possible integral curvature values. Under a mild integrability condition which rules out harmonic non-radial behavior near infinity, we prove that
u
is radially symmetric and decreasing in the following categories: (1)
K
is decreasing,
u
a classical solution, and the integral curvature of the surface is above critical; (2)
K
is decreasing,
u
a classical solution, the integral curvature of the surface is critical, and the surface satisfies an additional integrability condition which is mildly stronger than finite integral curvature; (3)
K
is non-positive. In categories 1 and 2,
K
is allowed to diverge logarithmically or as power law to
−∞
at spatial infinity. Examples of nonradial solutions which violate one or more of our conditions are discussed as well. In particular, for non-positive and non-negative
K
that satisfy appropriate integrability conditions and otherwise are fairly arbitrary, we introduce probabilistic methods to construct surfaces with finite integral curvature and entire harmonic asymptotics at infinity. For radial symmetric