We consider the problem of finding on a given Euclidean domainof dimension n≥3 a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form f(λ(−A))=1.This generalizes a proble...We consider the problem of finding on a given Euclidean domainof dimension n≥3 a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form f(λ(−A))=1.This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature.We prove the existence and uniqueness of such metric when the boundary δΩ is a smooth bounded hypersurface(of codimension one).When δΩ contains a compact smooth submanifold ∑ of higher codimension with δΩ\∑ being compact,we also give a‘sharp’condition for the divergence to infinity of the conformal factor near ∑ in terms of the codimension.展开更多
In this paper, we classify Mobius invariant differential operators of second orderin two-dimensional Euclidean space, and establish a Liouville type theorem forgeneral Mobius invariant elliptic equations. The equation...In this paper, we classify Mobius invariant differential operators of second orderin two-dimensional Euclidean space, and establish a Liouville type theorem forgeneral Mobius invariant elliptic equations. The equationsare naturally associ-ated with a continuous family of convex cones Γ_(p) in R^(2), with parameter p∈[1,2],joining the half plane Γ_(1) := {(λ_(1),λ_(2)) : λ_(1)+λ_(2)> 0} and the first quadrant Γ_(2) := {(λ_(1),λ_(2)) : λ_(1),λ_(2)> 0}. Chen and C. M. Li established in 1991 a Liouvilletype theorem corresponding to Γ_(1) under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville typetheorem we establish in this paper for Γ_(p),1 < p ≤ 2, does not require any additionalassumption on the solution as for Γ_(1). This is reminiscent of the I iouville type theo-rems in dimensions n≥3 established by Caffarelli, Gidas and Spruck in 1989 andby A.B. Li and Y. Y. Li in 2003-2005, where no additional assumption was neededeither. On the other hand, there is a striking new phenomena in dimension n=2 that Γ_(p) ,for p=1 is a sharp dividing line for such uniqueness result to hold without anyfurther assumption on the solution. In dimensions n≥3, there is no such dividing line.展开更多
基金supported by Spanish government Grants MTM2014-52402-C3-1-P and MTM2017-85757-Pthe BBVA Foundation Grant for Researchers and Cultural Creators 2016supported by NSF Grant DMS-1501004.
文摘We consider the problem of finding on a given Euclidean domainof dimension n≥3 a complete conformally flat metric whose Schouten curvature A satisfies some equations of the form f(λ(−A))=1.This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature.We prove the existence and uniqueness of such metric when the boundary δΩ is a smooth bounded hypersurface(of codimension one).When δΩ contains a compact smooth submanifold ∑ of higher codimension with δΩ\∑ being compact,we also give a‘sharp’condition for the divergence to infinity of the conformal factor near ∑ in terms of the codimension.
基金Yanyan Li’s research was partially supported by NSF Grants DMS-1501004,DMS-2000261,and Simons Fellows Award 677077Han Lu’s research was partially supported by NSF Grants DMS-1501004,DMS-2000261Siyuan Lu’s research was partially supported by NSERC Discovery Grant.
文摘In this paper, we classify Mobius invariant differential operators of second orderin two-dimensional Euclidean space, and establish a Liouville type theorem forgeneral Mobius invariant elliptic equations. The equationsare naturally associ-ated with a continuous family of convex cones Γ_(p) in R^(2), with parameter p∈[1,2],joining the half plane Γ_(1) := {(λ_(1),λ_(2)) : λ_(1)+λ_(2)> 0} and the first quadrant Γ_(2) := {(λ_(1),λ_(2)) : λ_(1),λ_(2)> 0}. Chen and C. M. Li established in 1991 a Liouvilletype theorem corresponding to Γ_(1) under an integrability assumption on the solution. The uniqueness result does not hold without this assumption. The Liouville typetheorem we establish in this paper for Γ_(p),1 < p ≤ 2, does not require any additionalassumption on the solution as for Γ_(1). This is reminiscent of the I iouville type theo-rems in dimensions n≥3 established by Caffarelli, Gidas and Spruck in 1989 andby A.B. Li and Y. Y. Li in 2003-2005, where no additional assumption was neededeither. On the other hand, there is a striking new phenomena in dimension n=2 that Γ_(p) ,for p=1 is a sharp dividing line for such uniqueness result to hold without anyfurther assumption on the solution. In dimensions n≥3, there is no such dividing line.
