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Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle’s results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander.

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_+) .

The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow on a closed (compact without boundary) manifold [Formula: see text]: [Formula: see text] In this paper, we prove the corresponding results on manifolds with boundary. More precisely, given a compact manifold [Formula: see text] with smooth boundary [Formula: see text], we study the convergence rate of the Yamabe flow with boundary: [Formula: see text] and the conformal mean curvature flow: [Formula: see text]

Let [Formula: see text] be a Riemannian minimal foliation. The transverse Yamabe problem is to find a metric [Formula: see text] in the basic conformal class of [Formula: see text] such that the transverse scalar curvature of [Formula: see text] is constant. We first study the uniqueness of the solutions of the transverse Yamabe problem. As a generalization of the transverse Yamabe problem, we study the problem of prescribing transverse scalar curvature by using geometric flow. We then prove a version of conformal Schwarz lemma on [Formula: see text]. Finally, we consider the transverse Yamabe soliton, which is the self-similar solution of the transverse Yamabe flow.

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.

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.

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.

Suppose $$M_{1}$$ and $$M_{2}$$ are 3-dimensional closed (compact without boundary) CR manifolds with positive CR Yamabe constant. In this note, we show that the connected sum of $$M_{1}$$ and $$M_{2}$$ also admits a CR structure with positive CR Yamabe constant.

Let $$(X,\omega _0)$$ be a compact complex manifold of complex dimension n endowed with a Hermitian metric $$\omega _0$$ . The Chern–Yamabe problem is to find a conformal metric of $$\omega _0$$ such that its Chern scalar curvature is constant. As a generalization of the Chern–Yamabe problem, we study the problem of prescribing Chern scalar curvature. We then estimate the first nonzero eigenvalue of Hodge–de Rham Laplacian of $$(X,\omega _0)$$ . On the other hand, we prove a version of conformal Schwarz lemma on $$(X,\omega _0)$$ . All these are achieved by using geometric flows related to the Chern–Yamabe flow. Finally, we prove the backwards uniqueness of the Chern–Yamabe flow.

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 note, we consider the problem of prescribing \begin{document}$ \overline{Q}' $\end{document} -curvature on a three-dimensional pseudohermitian manifold. Given a positive CR pluriharmonic function \begin{document}$ f $\end{document} , we construct a contact form on the three-dimensional pseudo-Einstein manifold with \begin{document}$ \overline{Q}' $\end{document} -curvature being equal to \begin{document}$ f $\end{document} , under some natural positivity conditions. On the other hand, we prove a Kazdan-Warner type identity for the problem of prescribing \begin{document}$ \overline{Q}' $\end{document} -curvature on the standard CR three sphere.

On a compact strictly pseudoconvex CR manifold $(M,\th)$, we consider the CR Yamabe constant of its infinite conformal covering. By using the maximum principles, we then prove a uniqueness theorem for the CR Yamabe flow on a complete noncompact CR manifold. Finally we obtain some properties of the CR Yamabe soliton on complete noncompact CR manifolds., Comment: To appear in NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS

In this paper, we study some global properties of curves in the Heisenberg group $$H_{1}$$. In particular, we obtain Fenchel-type theorem and Fary–Milnor type theorem, together with Bray–Jauregui type theorem. We also prove the fundamental theorem of curves in the standard CR 3-sphere $$S^{3}$$. As an application, we classify some horizontally regular curves in $$S^{3}$$.

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.

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold $M$ of real dimension $2n+1$. We prove convergence of the CR Yamabe flow when $n=1$ or $M$ is spherical., Accepted by Mathematische Annalen

In this paper, we prove the following version of conformal CR positive mass theorem: Suppose that \((N, J,\theta )\) and \((N, J,\hat{\theta }=e^{2f}\theta )\) are three-dimensional asymptotically flat pseudohermitian manifolds such that their Tanaka-Webster curvatures satisfy \(e^{2f}\hat{R}-R\ge 0.\) Then the p-mass of \((N, J, \theta )\) and \((N, J, \hat{\theta })\) satisfy \( m(J, \hat{\theta })-m(J, \theta )\ge 0, \) and equality holds if and only if \(\hat{\theta }=\theta \). We also prove that the p-mass is independent of the choice of the sequence of coordinates spheres.

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.

Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

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.

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.

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 study the Ricci–Bourguignon flow of all locally homogenous geometries on closed three-dimensional manifolds. We also consider the evolution of the Yamabe constant under the Ricci–Bourguignon flow. Finally, we prove some results for the Bach-flat shrinking gradient soliton to the Ricci–Bourguignon flow.

Suppose $(M,g_0)$ is a compact Riemannian manifold without boundary of dimension $n\geq 3$. Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of $g_0$ with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold., This is the full and detailed version. Accepted by Annals of Global Analysis and Geometry

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

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is $f$, when $f$ possesses certain reflection or rotation symmetry., Comment: 12 pages. Now published in Proc. Roy. Soc. Edinburgh Sect. A. Some typos are corrected

Let M be a compact n-dimensional hypersurface in the space form $${F^{n+1}(c)}$$ where n = 3 or 4. We prove that if the scalar curvature of M is positive and the integral of the length of the second fundamental form of M satisfies certain inequality, then M must be diffeomorphic to the spherical space form. On the other hand, we prove that if M is homeomorphic to 2-dimensional sphere $${\mathbb{S}^2}$$ and f 1, f 2, f 3 are first eigenfunctions of M such that $${\sum\nolimits_{i=1}^3|\nabla f_{i}|^2}$$ is a constant, then M is isometric to a sphere with constant curvature.

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

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal invariant $Y(M,\partial M), Theorem 1.1 improved, one reference added, typos fixed

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]?.