# Ngô Quốc Anh

### Preprints

Selected Preprints and working papers

2017

1. Q.A. Ngo, H. Zhang, Global existence and convergence of Q-curvature flows on manifolds of even dimension, submitted.
2. T.V. Duoc, Q.A. Ngo, Exact growth at infinity for radial solutions of $\Delta^3 u + u^{-q} = 0$ in $\mathbf R^3$, submitted.
3. Q.A. Ngo, V.H. Nguyen, Moser-Trudinger inequality involving the first non-zero Neumann eigenvalue with mean value zero in $\mathbf R^2$, preprint.
4. Pointwise, preprint.
5. Veron-Souplet pointwise, in preparation.
6. T.V. Duoc, Q.A. Ngo, in preparation.
7. Q.A. Ngo, Sobolev inequality involving the scalar and mean curvatures on Riemannian manifolds, work in progress.
8. Q.A. Ngo, H. Zhang, Global existence and convergence of Yamabe flow with boundary, work in progress.
9. Q.A. Ngo, V.H. Nguyen, N.T. Tai, works in progress.
10. Q.A. Ngo, V.H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality: The case of the Heisenberg group, work in progress.

2016

1. Q.A. Ngo, On a non-existence result for Einstein constraint-type systems with non-positive Yamabe constant, unpublished notes, 07 pages,
Abstract: We extend a recent non-existence result for an Einstein constraint-type system due to Dahl-Gicquaud-Humbert (Class. Quantum Grav. 30, 075004) from non-vanishing, non-positive metrics to metrics with non-positive Yamabe constant.
2. L. Hong, Q.A. Ngo, X. Xu, On the scalar curvature functions on closed manifolds with positive Yamabe constant, in preparation.
Abstract: In this paper, we prove that on a given closed Riemannian manifold of dimension $n \geqslant 3$ with positive Yamabe constant which is not conformally equivalent to the unit sphere, any smooth function is the scalar curvature of some metric in the given conformal class if and only if the function is positive somewhere.
3. Q.A. Ngo, V.H. Nguyen, Sharp Adams-Moser-Trudinger type inequalities in the hyperbolic space, preprint.
Abstract: The purpose of this paper is to establish some Adams-Moser-Trudinger inequalities, which are the borderline cases of the Sobolev embedding, in the hyperbolic space $\mathbb H^n$. First, we prove a sharp Adams inequality of order two with the exact growth condition in $\mathbb H^n$. Then we use it to derive a sharp Adams-type inequality and an Adachi-Tanaka-type inequality. We also prove a sharp Adams-type inequality with Navier boundary condition on any bounded domain of $\mathbb H^n$, which generalizes the result of Tarsi to the setting of hyperbolic spaces. Finally, we establish a Lions-type lemma and an improved Adams-type inequality in the spirit of Lions in $\mathbb H^n$. Our proofs rely on the symmetrization method extended to hyperbolic spaces.
4. Q.A. Ngo, V.H. Nguyen, Sharp constant for Poincaré-type inequalities in the hyperbolic space, submitted.
5. Q.A. Ngo, V.H. Nguyen, H. Zhang, Sobolev trace inequality involving the scalar and mean curvatures on Riemannian manifolds, preprint, 22 pages.
6. N.T. Dung, N.N. Khanh, Q.A. Ngo, Gradient estimates for some $f$-heat equations driven by Lichnerowicz’s equation on complete smooth metric measure spaces, submitted.
7. Q.A. Ngo, Classification of entire solutions of $(-\Delta )^Nu+u^{-(4N-1)}=0$ with exact linear growth at infinity in $\mathbf R^{2N-1}$, preprint.
8. Q.A. Ngo, Notes on prescribing curvature flow on surfaces with sign-changing candidate, in preparation.  [No longer interested in.]
9. Q.A. Ngo, Remarks on sufficient conditions for the prescribed scalar curvature problem, in preparation.  [No longer interested in.]
10. Quantum field theory, if I have time.

2015

1. Q.A. Ngo, Classification of solutions for a system of integral equations with negative exponents via the method of moving spheres, unpublished notes, 15 pages. (Now incorporated into this paper arXiv.org.)
Abstract: The main objective of the present note is to study positive solutions of the following interesting system of integral equations in $\mathbb R^n$

$\displaystyle \left\{ \begin{array}{lcl} u(x) = \displaystyle \int_{\mathbb R^n} { |x-y|^p v(y)^{-q} dy},\\v(x) = \displaystyle \int_{\mathbb R^n} { |x-y|^p u(y)^{-q} dy},\end{array} \right. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (\star)$

with $p, q >0$ and $n \geqslant 1$. Under the nonnegative Lebesgue measurability condition for solutions $(u,v)$ of ($\star$), we prove that $p q = p+2n$ must hold and that $u$ and $v$ are radially symmetric and monotone decreasing about some point. To prove this, we introduce an integral form of the method of moving spheres for systems to tackle ($\star$). As far as we know, this is the first attempt to use the integral form of the method of moving spheres for systems.

2. Q.A. Ngo, X. Xu, Slow convergence of the Gaussian curvature flow on closed Riemannian surfaces with vanishing Euler characteristic, preprint, 20 pages.
Abstract: In this note, we describe an alternative approach to solve the prescribed Gaussian curvature problem with vanishing Euler characteristic on closed Riemannian surfaces via a negative gradient curvature flow. The rate of the convergence of the flow is also studied in this note.
3. Q.A. Ngo, X. Xu, A flow approach to the prescribed scalar curvature problem with vanishing Yamabe invariant, preprint, 70 pages (final version, revision). (The result has been announced several times before, for example, here and here.)
Abstract: In this paper, we describe an alternative approach to some quantitative results of Escobar and Schoen for metrics of prescribed scalar curvature on closed Riemannian manifolds with vanishing Yamabe invariant via a negative gradient curvature flow. The main result is to remove the so-called flatness condition proposed by Escobar and Schoen; hence answering a conjecture by Kadzan and Warner affirmatively.

2011

1. Q.A. Ngo, X. Xu, Liouville type result for smooth positive solutions of the Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds. (Now incorporated into an paper hosted by .)

2009

1. Q.A. Ngo, On the uniqueness of eigenfunctions corresponding to the least eigenvalues of quasilinear eigenvalue problems.

2008

1. Q.A. Ngo, Multiplicity of solutions for a problem of $p$-Laplacian type with concave and convex nonlinearities.

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Last update: February 10, 2014.