Completion of R2 with a Conformal Metric as a Closed Surface

Changfeng Guiand Qinfeng Li

1 Department of Mathematics,The University of Texas at San Antonio,San Antonio,Texas 78249,USA

2 School of Mathematics,Hunan University,Changsha,Hunan 410082,China

Abstract.In this paper,we obtain some asymptotic behavior results for solutions to the prescribed Gaussian curvature equation.Moreover,we prove that under a conformal metric in R2,if the total Gaussian curvature is 4π,the conformal area of R2 is finite and the Gaussian curvature is bounded,then R2 is a compact C1,αsurface after completion at∞,for anyα∈(0,1).If the Gaussian curvature has a H¨older decay at infinity,then the completed surface is C2.For radial solutions,the same regularity holds if the Gaussian curvature has a limit at infinity.

Key Words:Gaussian curvature,conformal geometry,semilinear equations,entire solutions.

1 Introduction

In this paper,we consider the prescribed Gaussian curvature equation

where K satisfies

(1.1)is equivalent to that K is the Gaussian curvature of(R,eδ),whereδis the Euclidean metric,and hence(1.2)means that the total Gaussian curvature is finite.Anatural question is the following:

Question 1.1.If u is an entire Csolution to(1.1),then by assuming what conditions on K(x)can(R,eδ)be a Cclosed Riemannian surface after completion at∞?

Note that necessary conditions for this to be true include that

A natural question is the following:

Question 1.2.Are(1.3a)-(1.3c)sufficient to guarantee that(R,eδ)is a Cclosed Riemannian surface after completion at∞?

This question is related to a more general question in[6](Question 8.3)regarding the total area of Requipped with a conformal metric eδwith its Gaussian curvature bigger than 1,i.e.,with u being a super solution of(1.1).

Notice that(1.3c)implies that

For the convenience of later discussion,we define

In the following,(1.3b)and(1.4)will serve as fundamental assumptions.

is a Cfunction near x=0,and limh(x)>0,which means the metric at∞is nondegenerate.Hence we need to closely study the asymptotic behavior of u at∞.

We are mostly interested in the caseλ=2 since it corresponds to(1.3a).

Our first result concerns the asymptotic behavior of u and its partial derivatives when|x|is large.

where Cis a constant,0<α<1∧(2λ−3),and Care constants given by

For generalλ>1,we have

for anyα∈(0,1∧(2λ−2)),and

In addition,if we further assume that for i=1,2,

Remark 1.1.(1.8)is already proved in[3]under essentially weaker conditions,and the proof is very technical.The new ingredient in our theorem above is that under more convenient but natural assumptions(1.3b)and(1.4),we have established more precise asymptotic behaviors of u and Du for various ranges ofλ.

As a consequence of Theorem 1.1,we state an answer to Questions 1.1,1.2 as follows.

Theorem 1.2.Assume u and K satisfy(1.1),(1.3a)and(1.3b),and we also assume that K is a H¨older continuous function and satisfies

Then(R,eδ)can be completed as a Cclosed(compact)surface.

We also study radial solutions to(1.1).First,we show that

Theorem 1.3.Let u be a radial solution to(1.1),and we also assume(1.3b)and(1.4),then for anyλ>1,when r=|x|is large,we have that

and that

If K further satisfies(1.3c),then

As a consequence of Theorem 1.3,it turns out that the answer to Question 1.2 for radial solutions is positive,without(1.12)being assumed.

Corollary 1.1.If u is a radial solution to(1.1),then(R,eδ)can be completed at∞such that it becomes a Ccompact Riemannian surface,if and only if(1.3a)-(1.3c)are satisfied.

We organize the notes as follows.In Section 2,we prove Theorem 1.1 and Theorem 1.2.In Section 3,we prove Theorem 1.3 and Corollary 1.1.

2 Asymptotic behavior of general solutions to(1.1)

In this section,we study asymptotic behavior of solutions to(1.1),and we will prove Theorem 1.1 and Theorem 1.2.

Proof of Theorem 1.1.Let

Then as in[2],∆w(x)=K(x)e,and

By[1],(1.3b)and(1.4)imply that u is bounded from above.Then by the argument of[2],we have that u+w≡C for some constant C.

Let

then v(x)satisfies

where

Since u+w≡C,and by the asymptotic behavior of w,we have that for any∊>0,

where

where Cis a constant,A is a constant vector,andα∈(0,1∧(2λ−3)).Hence if|x|is large,we have

Next,we will prove(1.6).Let

and in the following,we will use C to denote various constants,possibly depending on λand‖K‖.

By the differential property of Newtonian potential,we have

Since u+w is a constant,for i=1,2,by the definition ofλwe have

where Ais given by

Now in order to figure out the constant vector A in(2.3),we compute

Since

it follows that

By(2.6),and since from(2.4)we have

it follows that

Since

we have

where we have used

by Dominated Convergence Theorem we have

Hence

Similarly,

Therefore,we have explicitly computed the constant vector A in(2.3)and(2.4),namely

This proves(1.6).Now we consider

Since

Also,since when

we have

It remains to estimate

Since

Using that|x·y|≤|x||y|,we similarly have that

Similarly,

Based on the estimates on I I,I I I,I,I,Iabove,we therefore have

This proves(1.9).

Using the exact estimate of|I|above,we have

Also,since

by the same estimate of Ias above,we have

Therefore,combing above,we have

This implies(1.11).

Proof of Theorem 1.2.We rewrite(1.5)as

h∈C(B(0)) when λ=2.

Since h(0)>0,the proof is completed.

Also from Theorem 1.1,we have

Corollary 2.1.Let u be a solution to(1.1),λ≥2 and(1.3b)holds.Moreover,if K is H¨older continuous and satisfies(1.12),then

where C,i=1,2 are constants given by(1.7)and Care also constants.

Remark 2.2.One can see from above that under the assumptions of Corollary 2.1,and if(1.10)is assumed,then

which is sharp in the sense that standard bubble solutions do have such asymptotic behavior.

3 Radial solutions to(1.1)

In this section,we study radial solutions to(1.1)and we will prove Theorem 1.3 and Corollary 1.1.

When K(x)is radially symmetric,under certain conditions it can be shown that a solution to(1.1)is also radially symmetric(see,e.g.,[3]Theorem 1.7 and[7]Theorem 5.2).

Before giving the proofs,we remark that by[4,Corollary 1.4 and Corollary 1.5],there always exists a Cradial solution to(1.1)if K(x)is H¨older continuous,radial,nonpositive near 0,and nonpositive near infinity(this last condition can be dropped if K further satisfies(1.12)).Therefore,one should not worry about the existence of solutions to(1.1)within our assumptions.

Proof of Theorem 1.3.If u is a radial solution to(1.1),then integrating(1.1)over B,and by the divergence theorem,we have

Hence

This proves(1.13).Moreover,(1.14)follows from(1.13)and Eq.(1.1),since

If(1.3c)is further satisfied,then it is easy to see from the above computations that(1.15a)and(1.15b)hold,since

where Cis the coefficient in(1.8).

Proof of Corollary 1.1.We simply write u(x)=u(|x|).The corollary follows from the better regularity theory for radial functions.Here we show the assertion directly.Note that by checking the proof of Theorem 1.1,we know that(1.15a)and(1.15b)are true if K satisfies(1.3a)and(1.3c).Hence as r→0,

where A,i=0,1,2,3 are constants.Since u is radial,

Hence

By direct computation,

Hence limh(r)=0,and thus L-Hospital’s Rule implies

Hence h(r)is Cat r=0.Also,applying(3.1a)-(3.1d),we have

Hence

Again,L-Hospital’s rule implies that

Hence h(r)is Cat r=0.

Also,one can check that A=4,A=2,A=−6,A=−8,and hence A(4 A+20 A+2 A)=−16=h(0).

has a singularity at r=0.Indeed,h(r)→∞as r→0whenλ<2,while h(0)=0 is degenerate whenλ>2.Furthermore,h(r)is not Cif 2<λ<5/2 and is not Cif 2<λ<3.The metrics correspond to the so-called conformal metrics with conical singularities on S.

The reader is referred to[8,9]for detailed discussions about surfaces with conical singularities.

Remark 3.1.Recently Dong Ye informed the authors that the answer to Question 1.2 should be negative,and a counter example can be constructed using the classic example of the nonexistence of a Csolution of the Poisson equation in the unit disc for a continuous but not Cdata(see,e.g.,[5,Exercise 4.9]).Indeed,one can choose a solution v to∆v=f in C(B{0})∩C(B),0<α<1 but v/∈C(B),and extend v to C(R{0})so that v(x)=−2 ln|x|for|x|sufficiently large.Define

u(x)=v(x/|x|)−2 ln|x|,

then u∈C(R).Let

K(x)=−e∆u,

it can be verified that(1.3a)-(1.3c)hold.However,the completion of the surface is only Cbut not C.

Acknowledgements

This research is partially supported by NSF grant DMS-1601885 and DMS-1901914.The authors would like to thank Dong Ye for the remark regarding the negative answer of Question 1.2.