ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 05 Feb 2023 11:33:10 GMT2023-02-05T11:33:10Z50281- Classification of solutions of higher order conformally invariant equationshttps://scholarbank.nus.edu.sg/handle/10635/102991Title: Classification of solutions of higher order conformally invariant equations
Authors: Wei, J.; Xu, X.
Mon, 01 Feb 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1029911999-02-01T00:00:00Z
- Erratum to: The scalar curvature flow on S n-perturbation theorem revisited (Invent Math, 10.1007/s00222-011-0335-6)https://scholarbank.nus.edu.sg/handle/10635/104680Title: Erratum to: The scalar curvature flow on S n-perturbation theorem revisited (Invent Math, 10.1007/s00222-011-0335-6)
Authors: Chen, X.; Xu, X.
Wed, 01 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1046802012-02-01T00:00:00Z
- Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in ℝ3https://scholarbank.nus.edu.sg/handle/10635/104427Title: Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in ℝ3
Authors: Wei, J.; Xu, X.
Abstract: Under some conditions on f (u), we show that for λ small and Ω R convex, the only solution to the elliptic equation Δu - λu + f (u) = 0 in Ω, with u > 0 in Ω and u/v = 0 on Ω, is constant.
Thu, 01 Sep 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044272005-09-01T00:00:00Z
- Uniqueness theorem for the entire positive solutions of biharmonic equations in ℝnhttps://scholarbank.nus.edu.sg/handle/10635/104436Title: Uniqueness theorem for the entire positive solutions of biharmonic equations in ℝn
Authors: Xu, X.
Abstract: The entire positive solutions of a conformally invariant biharmonic equation in ℝn will be classified using the method of moving spheres. As a byproduct, one also shows that any entire non-negative solution of the equation Δ2u = up with 1 ≤ p < (n + 4)/(n - 4) with n ≥ 2 is zero.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044362000-01-01T00:00:00Z
- On conformally invariant equation (-Δ)pu -K(x) uN+2p/N-2p = 0 and Its Generalizationshttps://scholarbank.nus.edu.sg/handle/10635/131457Title: On conformally invariant equation (-Δ)pu -K(x) uN+2p/N-2p = 0 and Its Generalizations
Authors: Lu, G.; Wei, J.; Xu, X.
Abstract: We consider the question of existence and non-existence of positive entire solutions for conformally invariant equations involving polybarmonic operator. We obtain existence of infinitely many positive solutions if the potential decays sufficiently fast at infinity and the nonexistence of positive solutions if the potential grows too fast at infinity. We also establish a Kazdan-Warner type condition for non-existence of solutions decaying at infinity.
Mon, 01 Jan 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1314572001-01-01T00:00:00Z
- Integral estimates of conformal metricshttps://scholarbank.nus.edu.sg/handle/10635/103431Title: Integral estimates of conformal metrics
Authors: Xu, X.
Abstract: In this article we show that there exists a rational number μ, depending only on the dimension n (≥ 5) of the manifold such that the μth-power of the conformal factor is bounded in H2 2 norm in terms of volume bound and the square norm bound of the scalar curvature of the conformal metrics. Some applications are also given. © 1996 American Mathematical Society.
Mon, 01 Jan 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034311996-01-01T00:00:00Z
- On the existence of extremal metricshttps://scholarbank.nus.edu.sg/handle/10635/103795Title: On the existence of extremal metrics
Authors: Xu, X.
Abstract: We study the well known variational problem proposed by Calabi: Minimize the functional ∫M s2 gdvg among all metrics in a given Kahler class. We are able to establish the existence of the extremal when the closed Riemann surface has genus different from zero. We have also given a different proof of the result originally proved by Calabi that: On a closed Riemann surface, the extremal metric has constant scalar curvature on a closed Riemann surface, the extremal metric has constant scalar curvature, which originally is proved by Calabi.
Sat, 01 Jun 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037951996-06-01T00:00:00Z
- Positivity of Paneitz Operatorshttps://scholarbank.nus.edu.sg/handle/10635/103961Title: Positivity of Paneitz Operators
Authors: Xu, X.; Yang, P.C.
Abstract: In this paper, we give two results concerning the positivity property of the Paneitz operator- a fourth order conformally covariant elliptic operator. We prove that the Paneitz operator is positive for a compact Riemannian manifold without boundary of dimension at least six if it has positve scalar curvature as well as nonnegative Q-curvature. We also show that the positivity of the Paneitz operator is preserved in dimensions greater than four in taking a connected sum.
Mon, 01 Jan 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039612001-01-01T00:00:00Z
- Q-curvature flow on the standard sphere of even dimensionhttps://scholarbank.nus.edu.sg/handle/10635/103992Title: Q-curvature flow on the standard sphere of even dimension
Authors: Chen, X.; Xu, X.
Abstract: Using a gradient flow approach initiated by S. Brendle, we generalize the existence theorem for the prescribing Q-curvature equation on S2 (Gauss curvature) by M. Struwe (2005) [14] and on S4 by Malchiodi and Struwe (2006) [12] to Sn for all even n with the similar assumption on the prescribed curvature candidate f. © 2011 Elsevier Inc.
Mon, 15 Aug 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039922011-08-15T00:00:00Z
- Constant mean curvature spheres in Riemannian manifoldshttps://scholarbank.nus.edu.sg/handle/10635/103045Title: Constant mean curvature spheres in Riemannian manifolds
Authors: Pacard, F.; Xu, X.
Abstract: We prove the existence of embedded spheres with large constant mean curvature in any compact Riemannian manifold (M, g). This result partially generalizes a result of R. Ye which handles the case where the scalar curvature function of the ambient manifold (M, g) has non-degenerate critical points. © 2008 Springer-Verlag.
Sun, 01 Mar 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030452009-03-01T00:00:00Z
- Conformal energy in four dimensionhttps://scholarbank.nus.edu.sg/handle/10635/103039Title: Conformal energy in four dimension
Authors: Xu, X.; Yang, P.C.
Abstract: The conformal energy for 4-manifolds using the Paneitz operator is introduced in this article. The conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume. We also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator.
Tue, 01 Jan 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030392002-01-01T00:00:00Z
- The complete hyper-surfaces with zero scalar curvature in ℝn+1https://scholarbank.nus.edu.sg/handle/10635/104266Title: The complete hyper-surfaces with zero scalar curvature in ℝn+1
Authors: Yaowen, L.; Xingwang, X.; Jiuru, Z.
Abstract: Let Mn be a complete and noncompact hyper-surface immersed in Rn+1. We should show that if M is of finite total curvature and Ricci flat, then M turns out to be a hyperplane. Meanwhile, the hyper-surfaces with the vanishing scalar curvature is also considered in this paper. It can be shown that if the total curvature is sufficiently small, then by refined Kato's inequality, conformal flatness and flatness are equivalent in some sense. And those results should be compared with Hartman and Nirenberg's similar results with flat curvature assumption. © 2013 The Author(s).
Sun, 01 Dec 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1042662013-12-01T00:00:00Z
- Classification of solutions of certain fourth-order nonlinear elliptic equations in R{double-struck}4https://scholarbank.nus.edu.sg/handle/10635/102990Title: Classification of solutions of certain fourth-order nonlinear elliptic equations in R{double-struck}4
Authors: Xingwang, X.U.
Abstract: We consider the uniqueness of solutions of the equation Δ2 u = eu in fourdimensional Euclidean space. Our main result is that the solutions are all classical ones, provided that the energy of the solutions is finite and the diffusion of the solutions decays to zero at infinity. The method we used in this paper is known as the method of moving spheres.
Thu, 01 Jun 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1029902006-06-01T00:00:00Z
- The scalar curvature flow on S n-perturbation theorem revisitedhttps://scholarbank.nus.edu.sg/handle/10635/104345Title: The scalar curvature flow on S n-perturbation theorem revisited
Authors: Chen, X.; Xu, X.
Abstract: For the problem of finding a geometry on S n, for n ≥ 3, with a prescribed scalar curvature, there is a well-known result which is called the perturbation theorem; it is due to Chang and Yang (Duke Math. J. 64, 27-69, 1991). Their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm. It is important to know how large that difference in sup norm can possibly be. Here we consider prescribing scalar curvature problem using the scalar curvature flow. For simplicity, we assume that the given curvature candidate f is a smooth positive Morse function which is non-degenerate in the sense that,. {pipe} Δ f {pipe} 2 S n f) 2 ≠ 0 For δ n = 2 2/n when n=3,4 and δ n=2 2/(n-2) for n ≥ 5, we show that if maxS n f/ minS n f < δ n, then f can be realized as the scalar curvature of some conformal metric provided that the degree counting condition holds for f. This shows that the best possible difference in the sup norm is n(n-1)(δ n-1)/(δ n+1). © 2011 Springer-Verlag.
Wed, 01 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1043452012-02-01T00:00:00Z
- On the classification of stable solutions to biharmonic problems in large dimensionshttps://scholarbank.nus.edu.sg/handle/10635/103775Title: On the classification of stable solutions to biharmonic problems in large dimensions
Authors: Wei, J.; Xu, X.; Yang, W.
Abstract: We give a new bound on the exponent for nonexistence of stable solutions to the biharmonic problem Δ2u=up in Rn, where u > 0, p > 1, and n ≥ 20.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1037752013-01-01T00:00:00Z
- Uniqueness and non-existence theorems for conformally invariant equationshttps://scholarbank.nus.edu.sg/handle/10635/104428Title: Uniqueness and non-existence theorems for conformally invariant equations
Authors: Xu, X.
Abstract: By using the equivalent integral form for the Q-curvature equation, we generalize the well-known non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q-curvature equation. Somehow, we introduce a new approach to Q-curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q = 1 solutions on Sn as well as on Rn with necessary growth rate assumption. © 2004 Elsevier Inc. All rights reserved.
Sun, 01 May 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044282005-05-01T00:00:00Z
- On a fourth order equation in 3-Dhttps://scholarbank.nus.edu.sg/handle/10635/103672Title: On a fourth order equation in 3-D
Authors: Xu, X.; Yang, P.C.
Abstract: In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold. © EDP Sciences, SMAI 2002.
Sat, 01 Jun 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036722002-06-01T00:00:00Z
- On Conformal Deformations of Metrics onSnhttps://scholarbank.nus.edu.sg/handle/10635/103695Title: On Conformal Deformations of Metrics onSn
Authors: Wei, J.; Xu, X.
Abstract: OnSn, there is a naturally metric definednth order conformal invariant operatorPn. Associated with this operator is a so-calledQ-curvature quantity. When two metrics are pointwise conformally related, their associated operators, together with theirQ-curvatures, satisfy the natural differential equations. This paper is devoted to the question of which function can be aQ-curvature candidate. This is the so-calledprescribing Q-curvature problem. Our main result is that ifQis positive, nondegenerate and the naturally defined mapping associated withQhas nonzero degree, then our problem has a solution. This is the natural generalization of prescribing Gaussian curvature onS2intoSn. © 1998 Academic Press.
Sat, 01 Aug 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036951998-08-01T00:00:00Z
- Uniqueness theorem for integral equations and its applicationhttps://scholarbank.nus.edu.sg/handle/10635/104435Title: Uniqueness theorem for integral equations and its application
Authors: Xu, X.
Abstract: This paper is devoted to answering a question asked recently by Y. Li regarding geometrically interesting integral equations. The main result is to give a necessary and sufficient condition on the parameters so that the integral equation with parameters to be discussed in this paper have regular solutions. In the case such condition is satisfied, we will write down the exact solution. As its application of our method, we should show that the non-existence theory of the solutions of prescribed scalar curvature equation on Sn can be generalized to that of prescribed Branson-Paneitz Q-curvature equations on Sn. © 2007 Elsevier Inc. All rights reserved.
Fri, 01 Jun 2007 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044352007-06-01T00:00:00Z
- Least energy solutions of semilinear Neumann problems and asymptoticshttps://scholarbank.nus.edu.sg/handle/10635/103482Title: Least energy solutions of semilinear Neumann problems and asymptotics
Authors: Pan, X.-B.; Xu, X.
Abstract: The asymptotic behavior of the least energy solutions of a semilinear Neumann problem involving the critical Sobolev exponent on a bounded domain in R4 is studied. Our main concern is the effect of the geometry of the boundary and the critical index, as contained in the boundary conditions, on the existence and the asymptotic behavior of the solutions. © 1996 Academic Press, Inc.
Mon, 15 Jul 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1034821996-07-15T00:00:00Z
- Compactness of isospectral conformal metrics and isospectral potentials on a 4-manifoldhttps://scholarbank.nus.edu.sg/handle/10635/103011Title: Compactness of isospectral conformal metrics and isospectral potentials on a 4-manifold
Authors: Chen, R.; Xu, X.
Mon, 01 Jul 1996 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030111996-07-01T00:00:00Z
- Ricci flow with hyperbolic warped product metricshttps://scholarbank.nus.edu.sg/handle/10635/104059Title: Ricci flow with hyperbolic warped product metrics
Authors: Ma, L.; Xu, X.
Abstract: In this short note, we show an example that the negative curvature is preserved in the deformation of hyperbolic warped product metrics under Ricci flow. It is showed that the flow converges to a flat metric in the sense of Cheeger-Gromov as time going to infinity. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Fri, 01 Apr 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1040592011-04-01T00:00:00Z
- Prescribing Q-curvature problem on Snhttps://scholarbank.nus.edu.sg/handle/10635/103968Title: Prescribing Q-curvature problem on Sn
Authors: Wei, J.; Xu, X.
Abstract: Let Pn be the n-th order Paneitz operator on Sn, n ≥ 3. We consider the following prescribing Q-curvature problem on Sn:Pn u + (n - 1) ! = Q (x) en u on Sn, where Q is a smooth positive function on Sn satisfying the following non-degeneracy condition:(Δ Q)2 + | ∇ Q |2 ≠ 0 . Let G* : Sn → Rn + 1 be defined byG* (x) = (- Δ Q (x), ∇ Q (x)) . We show that if Q > 0 is non-degenerate and deg (frac(G*, | G* |), Sn) ≠ 0, then the above equation has a solution. When n is even, this has been established in our earlier work [J. Wei, X. Xu, On conformal deformation of metrics on Sn, J. Funct. Anal. 157 (1998) 292-325]. When n is odd, Pn becomes a pseudo-differential operator. Here we develop a unified approach to treat both even and odd cases. The key idea is to write it as an integral equation and use Liapunov-Schmidt reduction method. © 2009.
Thu, 01 Oct 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039682009-10-01T00:00:00Z
- Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null casehttps://scholarbank.nus.edu.sg/handle/10635/103966Title: Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null case
Authors: Ge, Y.; Xu, X.
Abstract: The main objective of this short note is to give a sufficient condition for a non constant function k to be Q curvature candidate for a conformal metric on a closed Riemannian manifold with the null Q-curvature. In contrast to the prescribed scalar curvature on the two-dimensional flat tori, the condition we provided is not necessary as some examples show. © 2007 Springer-Verlag.
Tue, 01 Apr 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1039662008-04-01T00:00:00Z
- Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-spacehttps://scholarbank.nus.edu.sg/handle/10635/104423Title: Uniform bound and a non-existence result for Lichnerowicz equation in the whole n-space
Authors: Ma, L.; Xu, X.
Abstract: In this Note, we give a uniform bound and a non-existence result for positive solutions to the Lichnerowicz equation in Rn. In particular, we show that positive smooth solutions to:Δ u + f (u) = 0, u > 0, in Rn wheref (u) = u- p - 1 - up - 1, are uniformly bounded. To cite this article: L. Ma, X. Xu, C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences.
Wed, 01 Jul 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1044232009-07-01T00:00:00Z
- Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifoldshttps://scholarbank.nus.edu.sg/handle/10635/103242Title: Existence results for the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
Authors: Ngô, Q.A.; Xu, X.
Abstract: This article mainly concerns with the non-existence, existence, and multiplicity results for positive solutions to the Einstein-scalar field Lichnerowicz equation on closed manifolds with a negative conformal-scalar field invariant. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques to the analysis of the Hamiltonian constraint equation, especially those cases when the prescribed scalar curvature-scalar field function may change sign. To our knowledge, such a problem remains open. © 2012 Elsevier Ltd.
Sun, 01 Jul 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032422012-07-01T00:00:00Z
- Nonlinear biharmonic equations with negative exponentshttps://scholarbank.nus.edu.sg/handle/10635/103616Title: Nonlinear biharmonic equations with negative exponents
Authors: Choi, Y.S.; Xu, X.
Abstract: In this paper, we study global positive C4 solutions of the geometrically interesting equation: Δ2 u + u- q = 0 with q > 0 in R3. We will establish several existence and non-existence theorems, including the classification result for q = 7 with exactly linear growth condition. © 2008 Elsevier Inc. All rights reserved.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1036162009-01-01T00:00:00Z
- Exact solutions of nonlinear conformally invariant integral equations in R3https://scholarbank.nus.edu.sg/handle/10635/103219Title: Exact solutions of nonlinear conformally invariant integral equations in R3
Authors: Xu, X.
Abstract: In this paper, we will study the entire positive C4 solutions of the geometrically interesting integral equation: u(x) = 1/8π ∫R3 x-y u-q(y) dy with 0 < q in R3. We will show that there are positive entire solutions which take the form u(x) = c(1+ x 2) 1/2 up to dilation and translations, only when q = 7. © 2004 Elsevier Inc. All rights reserved.
Sun, 10 Jul 2005 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1032192005-07-10T00:00:00Z