arXiv research

Connected sum of CR manifolds with positive CR Yamabe constant is possible.

Suppose $M_{1}$ and $M_{2}$ are two closed (compact with no boundary) spherical 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 spherical CR structure with positive CR Yamabe constant.

The study proves CR structures on specific three-manifolds are equivalent to standard structures.

Study CR Yamabe constant and CR structures on manifolds.

Study CR Yamabe constant, flow, and soliton on CR manifolds.

New approach linking CR Yamabe invariant to Sasaki structures.

Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-n…

Positive mass theorem and Yamabe equation on CR manifolds

The paper constructs contact forms on a sphere with constant Webster curvature and applies them to CR Yamabe problems.

We give an integral formula for the total $Q^\prime$-curvature of a three-dimensional CR manifold with positive CR Yamabe constant and nonnegative Paneitz operator. Our derivation includes a relationship between the Green's functions of the CR Laplacian and the $P^\prime$-operator.

Paper solves CR positive mass and Yamabe problems on weighted spaces.

In this paper, we show that the Webster scalar curvature of any compact CR Yamabe soliton must be constant.

CR Yamabe flow fails to converge on small deformations of the standard CR three-sphere.

Study of CR Yamabe problem on toric contact manifolds.

In this paper, we investigate the geometry and classification of three-dimensional CR Yamabe solitons. In the compact case, we show that any 3-dimensional CR Yamabe soliton must have constant Tanaka-Webster scalar curvature; we also obtain a classification under the assumption that their potential functions are in the …

We give a survey of our recent work describing a method which combines the Sasaki join construction with the admissible Kähler construction of to obtain new extremal and new constant scalar curvature Sasaki metrics, including Sasaki-Einstein metrics. The constant scalar curvature Sasaki metrics also provide explicit so…

Solves Neumann problem on CR manifold boundary.

The paper studies CR Yamabe solutions on Sasakian manifolds with nonnegative curvature.

Paper proves existence of minimum energy solutions in 5D contact spin manifolds.

Study proves curvature flow existence on CR manifolds.

Study on Hausdorff dimension of singular CR Yamabe problem.

We express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. Many solutions are given and discussed. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as …

We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem w…

The study classifies CR solitons based on $C_0$-positivity and negativity.

CR structure on S³ with non-compact solutions to CR Yamabe problem.

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.

Study convergence of Yamabe flow on singular spaces with positive constant.

In this paper, we study the CR Yamabe flow with zero CR Yamabe invariant. We use the CR Poincaré inequality and a Gagliardo-Nirenberg type interpolation inequality to show that this flow has long time solution and the solution converges to a contact form with flat pseudo-Hermitian scalar curvature exponentially.

Study shows long-term flow on special manifolds with positive Yamabe constant.

Paper finds conditions for non-Einstein relative Yamabe metrics.

Study rigidity on CR Yamabe equation on Sasakian manifolds.

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold endowed with the Webster metric hence formulate a version of the CR Yamabe problem for CR manifolds-with-boundary. This is shown to be a nonlinear subelliptic problem of variational origin.

We deform the contact form by the amount of the Tanaka-Webster curvature on a closed spherical $CR$ three-manifold. We show that if a contact form evolves with free torsion and positive Tanaka-Webster curvature as initial data, then a certain Harnack inequality for the Tanaka-Webster curvature holds.

Innovates contact structure invariant, non-decreasing under operations.

The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.

The paper proves solutions for Yamabe equations on manifolds with boundary.

The Yamabe flow converges to a specific function on compactified manifolds.

Paper proves nonnegativity of CR Paneitz operator for embeddable CR manifolds.

New Liouville-type results for CR Yamabe equation in Heisenberg group.

In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this sol…

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

We define an ADM-like mass, called p-mass, for an asymptotically flat pseudohermitian manifold. The p-mass for the blow-up of a compact pseudohermitian manifold (with no boundary) is identified with the first nontrivial coefficient in the expansion of the Green function for the CR Laplacian. We deduce an integral formu…

Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M.$ Let $$ denote a $CR$ developing map {equation*} Φ:\widetilde{M}\rightarrow S^{2n+1} {equation*}% where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+…

The paper proves conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.

Upper diameter bound for manifolds with positive scalar curvature.

Sharp decay found for solutions of a specific equation in Lie groups.

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solv…

For a closed Riemannian manifold $(M^m,g)$ of constant positive scalar curvature and any other closed Riemannian manifold $(N^n,h)$, we show that the limit of the Yamabe constants of the Riemannian products $(M\times N,g+rh)$ as $r$ goes to infinity is equal to the Yamabe constant of $(M^m \times R^n, [g+g_E])$ and is …