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Consider the classical Sobolev trace inequality ‖ ∇ φ ‖ L 2 ( R + n ) ≥ K ‖ φ ‖ L 2 ( n − 1 ) n − 2 ( ∂ R + n ) \begin{equation*} \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)} \geq K\|\varphi \|_{L^{\frac {2(n-1)}{n-2}}(\partial \mathbb {R}^n_+)} \end{equation*} for all φ ∈ W 0 1 , 2 ( R + n ) \varphi \in W^{1,2}_0(\mathbb {R}^n_+) , where K K is the best constant. Here, W 0 1 , 2 ( R + n ) W^{1,2}_0(\mathbb {R}^n_+) is the space obtained by taking the completion in the norm ‖ ∇ φ ‖ L 2 ( R + n ) \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)} of the set of all smooth functions with support contained in the closure of R + n \mathbb {R}^n_+ , and n ≥ 3 n\geq 3 . Let M \mathcal {M} be the set of functions for which we have equality in the Sobolev trace inequality above. In this note, we show that there is a positive constant α \alpha such that ‖ ∇ φ ‖ L 2 ( R + n ) 2 − K 2 ‖ φ ‖ L 2 ( n − 1 ) n − 2 ( ∂ R + n ) 2 ≥ α d ( φ , M ) 2 \begin{equation*} \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)}^2-K^2 \|\varphi \|_{L^{\frac {2(n-1)}{n-2}}(\partial \mathbb {R}^n_+)}^2\geq \alpha d(\varphi ,\mathcal {M})^2 \end{equation*} for all φ ∈ W 0 1 , 2 ( R + n ) \varphi \in W^{1,2}_0(\mathbb {R}^n_+) , where d d is the distance in the Sobolev space W 0 1 , 2 ( R + n ) W^{1,2}_0(\mathbb {R}^n_+) .

In this paper, we define the second generalized Yamabe invariant on manifolds with boundary. We prove some of its properties and study when the invariant is attained by some metric. In another direction, by using the conformal mean curvature flow, we prove a version of the conformal Schwarz lemma for manifolds with boundary.

In this paper, we study the prescribing Webster scalar curvature problem on strictly pseudoconvex CR manifolds of real dimension $$2n+1$$ . First, we study the CR Nirenberg problem and prove some existence results. Second, we provide the existence and a positive lower bound for a solution of the CR Yamabe problem with zero Webster scalar curvature on noncompact complete manifolds.

As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we study the equivariant Yamabe problem with boundary.

Using the flow method, we prove an existence result for the problem of prescribing the \begin{document}$ Q $\end{document} -curvature on the even dimensional sphere \begin{document}$ S^n $\end{document} . More precisely, we prove that there exists a conformal metric on \begin{document}$ S^n $\end{document} such that its \begin{document}$ Q $\end{document} -curvature is \begin{document}$ f $\end{document} , when \begin{document}$ f $\end{document} possesses certain symmetry.

On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.

In this paper, we consider the solitons to the fully nonlinear type Yamabe flows. We prove that any k -Yamabe soliton with positive k -curvature must have constant σ k -curvature on closed locally conformally flat manifolds. We also prove that any quotient Yamabe soliton with positive k and l curvatures must have constant σ k σ l -curvature on closed locally conformally flat manifolds. Similar results are also proved on compact locally conformally flat manifolds with totally geodesic boundary.

Suppose M is homeomorphic to 2-dimensional sphere and let \(f_i\), \(1\le i\le 3\), be the first eigenfunctions of the Laplacian on M. Cheng proved (Proc Am Math Soc 55(2):379–381, 1976) that if \(\sum _{i=1}^3 f_i^2\) is a constant, then M is isometric to a sphere of constant curvature. In view of the similarities between eigenvalues of the Laplacian and Steklov eigenvalue, we study eigenfuction of the first nonzero Steklov eigenvalue of a 2-dimensional compact manifold with boundary \(\partial M\). Suppose M is a domain equipped with the flat metric g, and let f be an eigenfunction of the first nonzero Steklov’s eigenvalue. We prove that if \(\nabla _g f\) is parallel along \(\partial M\), the Lie bracket of the vector field orthogonal to \(\nabla _g f\) and the tangent to \(\partial M\) is zero, and \(|\nabla _g f|^2\) is constant in M, then (M, g) is isometric to the disk equipped with the flat metric. We also prove that if the eigenfunctions \(f_1, f_2\) corresponding to the first nonzero Steklov eigenvalue satisfy that \(f_1^2+f_2^2\) is constant on \(\partial M\) and \(\varphi _*(g)\) is a Riemannian metric conformal to the flat metric of \(D^2\) where \(\varphi =(f_1,f_2)\), then (M, g) is isometric to the disk equipped with the flat metric. In another direction, we prove that a simply connected domain in the hyperbolic space \(\mathbb {H}^3\) such that its Steklov eigenvalues are the same as a geodesic ball must be isometric to the geodesic ball.

In this paper, we show that soliton to the fractional Yamabe flow must have constant fractional order curvature.

In this paper, we prove an existence result of prescribing Webster scalar curvature on the CR sphere in cases where the prescribed function exhibits reflection or rotation symmetry.

In this paper, we first study the fractional Yamabe solitons, which are the self-similar solutions to fractional Yamabe flow.We prove some rigidity results and Liouville type results for such solitons.We thenconsider the fractional Nirenberg problem:the problem of prescribing fractional order curvature on the sphere.More precisely, we prove that there exists a conformal metric on the unit sphere such that itsfractional order curvature is \begin{document}$ f $\end{document}, when \begin{document}$ f $\end{document} possesses certain reflection or rotation symmetry.

The Gauss–Bonnet–Chern mass was defined and studied by Ge, Wang, and Wu [Adv. Math. 266, 84–119 (2014)]. In this paper, we consider the evolution of Gauss–Bonnet–Chern mass along the Ricci flow and the Yamabe flow.

In this paper, we study the cross curvature soliton. We study the cross curvature soliton with a warped product structure. On the other hand, we show that the volume entropy is decreasing along the cross curvature flow.

In this paper, we first prove that any two conformal contact forms on a compact CR manifold that have the same pseudo-Hermitian Ricci curvature must be different by a constant. In another direction, we prove a CR analogue of the conformal Schwarz lemma of Riemannian geometry.

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f., To appear in Advances in Mathematics

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.

In this paper, we show that the Webster scalar curvature of any compact CR Yamabe soliton must be constant., To appear in J. Geom. Phys

In this paper, we study the problem of prescribing Q -curvature on the even-dimensional standard sphere S n as a given function f . Using the prescribed Q -curvature flow, which was initially studied by Malchiodi and Struwe on S 4 , we prove an existence result under suitable assumptions on f .

A longstanding problem of crossing number, Zarankiewicz's conjecture, asserts that the crossing number of the complete bipartite graph K"m","n is @[email protected][email protected][email protected][email protected][email protected][email protected][email protected]?, which is known only for [email protected]?6. It is natural to generalize Zarankiewicz conjecture and ask: What is the crossing number for the complete multipartite graph? In this paper, we prove the following lower bounds for the crossing number of K"1","m","n in terms of the crossing number of the complete bipartite graph: cr(K"1","m","n)>=cr(K"m"+"1","n"+"1)[email protected][email protected][email protected][email protected][email protected][email protected]?;cr(K"1","2"M","n)>=12(cr(K"2"M"+"1","n"+"2)+cr(K"2"M"+"1","n)-M(M+n-1)). As a corollary, we show that: 1.cr(K"1","m","n)>=0.8594Z(m+1,n+1)[email protected][email protected][email protected][email protected][email protected][email protected]?; 2.If Zarankiewicz's conjecture is true for m=2M+1, then cr(K"1","2"M","n)=M^[email protected][email protected][email protected][email protected][email protected][email protected]?; 3.cr(K"1","4","n)[email protected][email protected][email protected][email protected][email protected][email protected]?.

In [7], Wylie proved that a complete shrinking Ricci soliton has finite fundamental group. In this note, we show that if M is a complete non-expanding Ricci soliton, then under a technical assumption on the universal covering of M, any finitely generated subgroup of the fundamental group of M has polynomial growth.

Suppose that M is a compact orientable hypersurface embedded in a compact n-dimensional orientable Riemannian manifold N. Suppose that the Ricci curvature of N is bounded below by a positive constant k. We show that 2 λ 1 > k − ( n − 1 ) max M | H | where λ 1 is the first eigenvalue of the Laplacian of M and H is the mean curvature of M.

In this paper, we study the crossing number of the circulant graph C(3k+1;{1,k}) and prove that cr(C(3k+1;{1,k}))=k+1 for k>=3.

We show that the crossing number of K"4","n on the real projective plane is n32n-31+n3.We also give an estimate for the crossing number of K"m","n on the real projective plane.

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

In this paper, we show that the toroidal crossing number of K"4","n, K"1","3","n, K"2","2","n, K"1","1","2","n and K"1","1","1","1","n is @?n4@?[2n-4(1+@?n4@?)]. In addition, a new lower bound for the toroidal crossing number of K"m","n has been obtained. We also discuss about the crossing number of K"4","n in the general surfaces.

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567–3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H 3 , then the first eigenvalue of M satisfies 1 / 4 ⩽ λ ( M ) ⩽ 4 / 3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in H n + 1 where n ⩾ 3 such that its smooth Yamabe invariant is negative, then ( n − 1 ) / 4 ⩽ λ ( M ) ⩽ n 2 ( n − 2 ) / ( 7 n − 6 ) .