Q Curvature on a Class of 3‐Manifolds

Motivated by the strong maximum principle for the Paneitz operator in dimension 5 or higher found in a preprint by Gursky and Malchiodi and the calculation of the second variation of the Green's function pole's value on ??³ in our preprint, we study the Riemannian metric on 3‐manifolds with positive scalar and Q curvature. Among other things, we show it is always possible to find a constant Q curvature metric in the conformal class. Moreover, the Green's function is always negative away from the pole, and the pole's value vanishes if and only if the Riemannian manifold is conformal diffeomorphic to the standard ??³. Compactness of constant Q curvature metrics in a conformal class and the associated Sobolev inequality are also discussed. © 2016 Wiley Periodicals, Inc.     

Compactification of a class of conformally flat 4-manifold

In this paper we generalize Huber’s result on complete surfaces of finite total curvature. For complete locally conformally flat 4-manifolds of positive scalar curvature with Q curvature integrable, where Q is a variant of the Chern-Gauss-Bonnet integrand; we first derive the Cohn-Vossen inequality. We then establish finiteness of the topology. This allows us to provide conformal compactification of such manifolds. Oblatum 3-III-1999 & 18-II-2000?Published online: 8 May 2000     

Compactness for conformal metrics with constant Q curvature on locally conformally flat manifolds

In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n ≥ 5 and with Poincaré exponent less than , the set of conformal metrics of positive constant Q and positive scalar curvature is compact in the C∞ topology.     

On Fractional GJMS Operators

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet‐to‐Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli‐Silvestre extension for (?Δ)^γ when γ ? (0,1), and both a geometric interpretation and a curved analogue of the higher‐order extension found by R. Yang for (?Δ)^γ when γ > 1. We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincaré‐Einstein manifold, including an interpretation as a renormalized energy. Second, for γ ? (1,2), we show that if the scalar curvature and the fractional Q‐curvature Q_2γ of the boundary are nonnegative, then the fractional GJMS operator P_2γ is nonnegative. Third, by assuming additionally that Q_2γ is not identically zero, we show that P_2γ satisfies a strong maximum principle.© 2016 Wiley Periodicals, Inc.     

Non-compactness of the prescribed Q-curvature problem in large dimensions

Let (M, g) be a compact Riemannian manifold of dimension N ≥ 5 and Q _g be its Q curvature. The prescribed Q curvature problem is concerned with finding metric of constant Q curvature in the conformal class of g. This amounts to finding a positive solution to ${P_{g}(u)=c u^{\frac{N+4}{N-4}},\quad u > 0 \quad {\rm on} \; M}$ where P _g is the Paneitz operator. We show that for dimensions N ≥ 25, the set of all positive solutions to the prescribed Q curvature problem is non-compact.     

Isoperimetric inequality from the poisson equation via curvature

In this paper, we establish an isoperimetric inequality in a metric measure space via the Poisson equation. Let (X,d,μ) be a complete, pathwise connected metric space with locally Ahlfors Q‐regular measure, where Q > 1, that supports a local L²‐Poincaré inequality. We show that, for the Poisson equation Δu = g, if the local L^∞‐norm of the gradient Du can be bounded by the Lorentz norm L^Q,1 of g, then we obtain an isoperimetric inequality and a Sobolev inequality in (X,d,μ) with optimal exponents. By assuming a suitable curvature lower bound, we establish such optimal bounds on $\||Du|\|_{L^\infty_{\rm loc}}$ . © 2011 Wiley Periodicals, Inc.     

A characterization of generalized poles of generalized Nevanlinna functions

Generalized poles of a generalized Nevanlinna function Q ∈ ??_κ (??) are defined in terms of the operator representation of Q . In this paper those generalized poles that are not of positive type and their degrees of non‐positivity are characterized analytically by means of pole cancellation functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prescribed Q-curvature problem on closed 4-Riemannian manifolds in the null case

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. The second author would like to thank Department of Mathematics, University of Paris XII for their invitation, hospitality and financial support during his visit. His research is partially supported by NUS research grant: R-146-000-077-112.     

Non‐isolated quasi‐degrees

We show that non‐isolated from below 2‐c.e. Q ‐degrees are dense in the structure of c.e. Q ‐degrees. We construct a 2‐c.e. Q ‐degree, which can't be isolated from below not only by c.e. Q ‐degrees, but by any Q ‐degree. We also prove that below any c.e. Q ‐degree there is a 2‐c.e. Q ‐degree, which is non‐isolated from below and from above (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gaussian curvature estimates for the convex level sets of p‐harmonic functions

We give a positive lower bound for the Gaussian curvature of the convex level sets of p‐harmonic functions with the Gaussian curvature of the boundary and the norm of the gradient on the boundary. Combining the deformation process, this estimate gives a new approach to studying the convexity of the level sets of the p‐harmonic function. © 2010 Wiley Periodicals, Inc.     

Incidence properties of cosets in loops

We study incidence properties among cosets of infinite loops, with emphasis on well‐structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a infinite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a infinite Bol loop S, can in some cases determine whether |S| divides |Q| for all infinite Bol loops Q with S?Q, and even whether there is a selection of left cosets of S that partitions Q. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop Q (in particular, in a commutative Moufang loop Q), two left cosets of S?Qare either disjoint or they intersect in a set whose cardinality equals that of some subloop of S.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

In this paper, we first show the global existence of the three-dimensionalCalabi flow on any closed 3-manifold with an arbitrary background metric g ₀. Second, we show the asymptotic convergence of a subsequence ofsolutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q < 0 or Q = 0 and Q > 0, up to conformal transformations. With itsapplication, we prove the existence of extremal metrics for quadraticfunctional of scalar curvature on a closed 3-manifold which is served asan extension of the Yamabe problem on closed manifolds. Moreover, theexistence of extremal metrics on complete noncompact 3-manifolds willdiscuss elsewhere.     

Conformal minimal two-spheres in Q 2

In this paper, we consider a conformal minimal immersion f from S ₂ into a hyperquadric Q ₂, and prove that its Gaussian curvature K and normal curvature K ^⊥ satisfy K + K ^⊥ = 4. We also show that the ellipse of curvature is a circle.     

Function cones and interpolation

We consider interpolation of operators acting on functions that belong to a given cone Q with the so‐called decomposition property. The set of all positive functions whose level sets are the level sets of a given function is the main example, and the cone of all decreasing functions is a particular case. As applications, we obtain conditions for the identity (E₀ ∩ Q,E₁ ∩ Q)_θ,p = (E₀,E₁)_θ,p ∩ Q and interpolation results for operators which are bounded when restricted to a given family of characteristic funcions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The k‐ary Montgomery modular inverse over nonbinary computers

This paper presents a k‐ary Montgomery modular inverse algorithm over nonbinary computers by using Sedjelmaci's right shift k‐ary greatest common divisor scheme. Over traditional binary computers, Kaliski's scheme can output Montgomery modular inverse Q ^{? 1}2ⁿ mod P, where P is coprime to Q and n is the bit length of P. Over k‐ary computers, our algorithm can discover the k‐ary Montgomery inverse Q ^{? 1}k^m mod P, where P, Q, and k are pairwise relatively prime positive integers and m = log _kP. In the worst case, the computational cost of our algorithm is O(m²)k‐ary digit operations. Copyright © 2013 John Wiley & Sons, Ltd.     

Factorizations of complete multipartite graphs into generalized cubes

For a positive integer d, the usual d‐dimensional cube Q_d is defined to be the graph (K₂)^d, the Cartesian product of d copies of K₂. We define the generalized cube Q(K_k, d) to be the graph (K_k)^d for positive integers d and k. We investigate the decomposition of the complete multipartite graph K into factors that are vertex‐disjoint unions of generalized cubes Q(K_k, d_i), where k is a power of a prime, n and j are positive integers with j ≤ n, and the d_i may be different in different factors. We also use these results to partially settle a problem of Kotzig on Q_d‐factorizations of K_n. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 144–150, 2000     

Existence of T*(3, 4,v)‐codes

A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q^*₄ be the set of all words of length 4 over Q. A T^*(3, 4, v)‐code over Q is a subset C^*? Q^*₄ such that every word of length 3 over Q occurs as a subword in exactly one word of C^*. Levenshtein has proved that a T^*(3, 4, vv)‐code exists for all even v. In this paper, the notion of a generalized candelabra t‐system is introduced and used to show that a T^*(3, 4, v)‐code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T^*(3,4, v)‐code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.     

On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary

In this paper we investigate the solvability of the Neumann problem (1.1) involving the critical Sobolev exponents on the right-hand side of the equation and in the boundary condition. It is assumed that the coefficients Q and P are smooth. We examine the common effect of the mean curvature of the boundary ∂Ω and the shape of the graph of the coefficients Q and P on the existence of solutions of problem (1.1).     

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method, we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary. These equations are fourth-order elliptic equations with critical Sobolev growth.