Remark on Gauss curvature equations on punctured disk
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Remark on Gauss curvature equations on punctured disk Yuxiang LI,
Hongyan TANG
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
c Higher Education Press 2020
Abstract We give a new argument on the classification of solutions of Gauss curvature equation on R2 , which was first proved by W. Chen and C. Li [Duke Math. J., 1991, 63(3): 615–622]. Our argument bases on the decomposition properties of the Gauss curvature equation on the punctured disk. Keywords Gauss curvature equation, singular point MSC 53C21, 58J05 1
Introduction
The classification of solutions of the equation 2u 2 −∆u = e , x ∈ R , Z e2u < +∞,
(1)
R2
has been solved by Chen and Li [2]. Using the moving frame, they proved the following result. Theorem 1 [2]
Let u solve (1). Then λ2 u = − log 1 + |x − x0 |2 + log λ, 4
where λ > 0. A complex-analytic proof of Theorem 1 can be found in [3]. In this paper, we will give a geometric proof of the above result. Roughly speaking, we consider R2 to be a complex chart of S2 \ {N }, where N is the north pole. Then g = e2u ((dx1 )2 + (dx2 )2 ) is a smooth metric with curvature 1 and finite volume, whenever u solves (1). It is well known that a smooth simple Received June 5, 2020; accepted July 21, 2020 Corresponding author: Yuxiang LI, E-mail: [email protected]
702
Yuxiang LI, Hongyan TANG
connected manifold with curvature 1 must be isometric to the standard sphere. Thus, once the smoothness of g on S2 is proved, Theorem 1 follows. For this reason, we will study the Gauss curvature equation −∆u = Ke2u on the 2-dimensional punctured disk D \ {0}, where Z 2 u ∈ C (D \ {0}), |K|e2u < +∞. D
Note that K is just the Gauss curvature of g = e2u ((dx1 )2 + (dx2 )2 ). Denote the disk with radius t and center 0 by Dt . We define Z ∂u 1 , λ(u) = lim t→0 2π ∂Dt ∂t
(2)
whose existence follows from the Green formula. We will prove the following result. Lemma 1
(i) If Z (1 + |K|)dµg < +∞, D
then λ(u) > −1, and there exists u0 ∈ ∩q∈(1,2) W 1,q (D1/2 ) such that u = u0 + λ(u) log |x|. Moreover, u solves the equation −∆u = Ke2u − 2πλ(u)δ0 in the sense of distribution. (ii) If Z
(1 + |K|p )dµg < +∞
D 0
for some p > 1 and λ(u) > −1, then u0 ∈ W 2,p for some p0 > 1. (iii) If K > 0 and Z (1 + K)dµg < +∞, D
then λ(u) > −1. Remark 1
In Lemma 1 (ii), λ > −1 is optimal. For example, when u = − log r − log log r,
the metric g = e2u (dx2 + dy 2 ) is complete on D1/2 \ {0} with K = −1,
vol(D, g) < +∞.
(3)
(4)
Remark on Gauss curvature equations on punctured disk
703
Then (4) and λ(u) = 0 imply u ∈ W 2,p (D). Hence, to prove Chen-Li’s result, it is enough to show λ(u) = 0 at the north pole. We should mention that λ(u) = 0 is not generally true, even when K = 1. For example, when |x|4 u = − log 1 + + log 2|x|, 4 we have K = 1,
vol(g) < +∞.
Fortunately, the solution of (1) has only one singularity on S2 . Then we are able to use Pohozaev inequality to get λ(u) = 0.
2
Decomposition lemma
By [1, Theorem 1] or [4, Theorem 2.2], taking a smooth sequence fk , whic
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