Uniqueness theorem for integral equations and its application

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.     

Q-curvature flow on the standard sphere of even dimension

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.     

Q-curvature flow on 4-manifolds with boundary

Given a compact four-dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero. Using integral methods, we prove global existence and convergence for the Q-curvature flow to a smooth metric conformal to g of prescribed Q-curvature, zero T-curvature and vanishing mean curvature under conformally invariant assumptions.     

Constant Q-curvature metrics in arbitrary dimension

Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order non-linear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.     

Conformal metrics with prescribed Q-curvature on S n

In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n????5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler?CHopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.     

Generic multiplicity for a scalar field equation on compact surfaces

We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.     

Singular solutions for the constant Q-curvature problem

This paper is devoted to the construction of weak solutions to the singular constant Q-curvature problem. We build on several tools developed in the last years. This is the first construction of singular metrics on closed manifolds of sufficiently large dimension with constant (positive) Q-curvature.     

Results of prescribing Q-curvature on S n

Using a flow approach, generalizing previous results from Chen and Xu [12] and Ho [20], an existence result of prescribing Q-curvature on the even-dimensional standard sphere S ⁿ as a given positive function f was obtained under suitable assumptions on the Morse index of f. In this note, we prove a more general existence result by assuming only that f is positive somewhere.     

Concentration-compactness phenomena in the higher order Liouville's equation

We investigate different concentration-compactness and blow-up phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in R2m, then that of a closed manifold and, finally, the particular case of the sphere S2m. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in R2m, blow-up phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have compactness.     

Priori estimates and asymptotic properties of solutions for some fractional order elliptic equations

In this paper, we obtain estimates for solutions for a class of fractional order elliptic equations in different domains and boundary conditions, and prove some regularity results. Then, we study the qualitative properties of solutions with prescribed Q-curvature.     

Holographic formula for Q-curvature

This paper derives an explicit formula for Branson's Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincaré metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

Constant Q-curvature metrics near the hyperbolic metric

Let (M,g)$(M,g)$ be a Poincaré–Einstein manifold with a smooth defining function. In this note, we prove that there are infinitely many asymptotically hyperbolic metrics with constant Q-curvature in the conformal class of an asymptotically hyperbolic metric close enough to g. These metrics are parametrized by the elements in the kernel of the linearized operator of the prescribed constant Q-curvature equation. A similar analysis is applied to a class of fourth order equations arising in spectral theory.     

The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds

We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

On Einstein Matsumoto metrics

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the (α,β)-metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n-dimensional (n≥3) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.     

GJMS operators and Q-curvature for conformal Codazzi structures

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersurfaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.     

On second grade fluids with vanishing viscosity

We consider in ℝⁿ (n = 2, 3) the equation of a second grade fluid with vanishing viscosity, also known as Camassa-Holm equation. We prove local existence and uniqueness of solutions for smooth initial data. We also give a blow-up condition which implies that the solution is global for n = 2. Finally, we prove the convergence of the solutions of second grade fluid equation to the solution of the Camassa-Holm equation as the viscosity tends to zero.