pseudo-Hermitian Manifolds Research Articles

Deformations of the Scalar Curvature of a Partially Integrable Pseudohermitian Manifold

Deformations of the Scalar Curvature of a Partially Integrable Pseudohermitian Manifold

Some Q-curvature Operators on Five-Dimensional Pseudohermitian Manifolds

We construct Q-curvature operators on d-closed (1, 1)-forms and on $$\overline{\partial }_b$$ -closed (0, 1)-forms on five-dimensional pseudohermitian manifolds. These closely related operators give rise to a new formula for the scalar Q-curvature. As applications, we give a cohomological characterization of CR five-manifolds which admit a Q-flat contact form, and we show that every closed, strictly pseudoconvex CR five-manifold with trivial first real Chern class admits a Q-flat contact form provided the Q-curvature operator on $$\overline{\partial }_b$$ -closed (0, 1)-forms is nonnegative.

Ancient Caloric Functions on Pseudohermitian Manifolds

Ancient Caloric Functions on Pseudohermitian Manifolds

Generalized Maximum Principles and Stochastic Completeness for Pseudo-Hermitian Manifolds

In this paper, the authors establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, they prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, they give some applications of these generalized maximum principles.

Prescribed Webster Scalar Curvatures on Compact Pseudo-Hermitian Manifolds

In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.

Liouville Theorems for Holomorphic Maps on Pseudo-Hermitian Manifolds

We prove some Liouville type results for generalized holomorphic maps in three classes: maps from pseudo-Hermitian manifolds to almost Hermitian manifolds, maps from almost Hermitian manifolds to pseudo-Hermitian manifolds and maps from pseudo-Hermitian manifolds to pseudo-Hermitian manifolds, assuming that the domains are compact. For instance, we show that any \((J,J^N)\) holomorphic map from a compact pseudo-Hermitian manifold M with nonnegative (resp. positive) pseudo-Hermitian sectional curvature to an almost Hermitian manifold N with negative (resp. nonpositive) holomorphic sectional curvature is constant. We also construct explicit almost CR structures on a complex vector bundle over an almost CR manifold.

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a K\"ahler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense of Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahler manifold of nonnegative holomorphic sectional curvature. Our approach is rather simple and uses a version of the Gauss equation for semi-isometric CR immersions of pseudohermitian manifolds into K\"ahler manifolds.

Curve shortening flow in a 3-dimensional pseudohermitian manifold

Curve shortening flow in a 3-dimensional pseudohermitian manifold

Uniqueness results on CR manifolds

In this paper we show the uniqueness of the constant solution to a general Yamabe-type equation on a compact pseudohermitian manifold. The Riemannian version was first proved by Bidaut-V eron and V eron. In another direction, we make use of the vanishing property of two eigenfunctions of the sub-Laplacian to conclude the nonvanishing of the first cohomology group of a compact pseudohermitian manifold. The corresponding version for the Rieman- nian case was first proved by Gichev.

Generalizations of the Theorems of Pappus-Guldin in the Heisenberg groups

In this paper, we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group $${\mathbb {H}}_1$$ , which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in $${\mathbb {H}}_1$$ and show that the classical result of Pappus-Guldin theorems for surface areas and volumes hold if the surfaces satisfy some geometric properties. Some examples are also provided, including the surfaces with constant p-mean curvatures.

GENERAL SCHWARZ LEMMAS BETWEEN PSEUDO-HERMITIAN MANIFOLDS AND HERMITIAN MANIFOLDS

<abstract> From the viewpoint of differential geometry, Schwarz lemmas of distance-decreasing type and volume-decreasing type can be obtained by the estimates of sum functions and product functions of all eigenvalues of holomorphic maps. This paper investigates general Schwarz lemmas by estimating partial sum functions and partial product functions of the eigenvalues of generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. </abstract>

Schwarz‐type lemmas for generalized holomorphic maps between pseudo‐Hermitian manifolds and Hermitian manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carath\'eodory pseudodistance on CR manifolds.

CR Gradient Estimate for the Positive Eigenfunction of Witten Sub-Laplacian

In this paper, under m-Bakry–Emery (or $$\infty $$ -Bakry–Emery) pseudohermitian Ricci curvature condition, we derive subgradient estimate for positive solutions of $$\Delta _{H,\phi } u(x)=-\lambda u(x)$$ on a weighted complete noncompact pseudohermitian manifold which satisfies the sub-Laplacian comparison property.

Schwarz Type Lemmas for Pseudo-Hermitian Manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the \emph{CR} maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the \emph{CR} maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the \emph{CR} hyperbolicity problem for pseudo-Hermitian manifolds.

On Li–Yau gradient estimate for sum of squares of vector fields up to higher step

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.

Gradient estimate of positive eigenfunctions of sub-Laplacian on complete pseudo-Hermitian manifolds

In this paper, we derive an explicit gradient estimate of positive solutions of Δbu=−λu on complete noncompact pseudo-Hermitian manifolds with bounded geometric conditions. As a consequence, we obtain an estimate of the greatest lower bound for the L2-spectrum of the sub-Laplacian operator. Moreover, we recapture the Liouville theorem of positive pseudo-harmonic functions on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.

CR-HARMONIC MAPS

We develop the notion of renormalized energy in Cauchy–Riemann (CR) geometry for maps from a strictly pseudoconvex pseudo-Hermitian manifold to a Riemannian manifold. This energy is a CR invariant functional whose critical points, which we call CR-harmonic maps, satisfy a CR covariant partial differential equation. The corresponding operator coincides on functions with the CR Paneitz operator.

On the Sharp Dimension Estimate of CR Holomorphic Functions in Sasakian Manifolds

Abstract This is the very 1st paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e., Sasakian manifold), which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.

CR-analogue of the Siu-∂\overline{∂}-formula and applications to the rigidity problem for pseudo-Hermitian harmonic maps

We give several versions of Siu’s ∂ ∂ ¯ \partial \overline {\partial } -formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold ( M 2 m + 1 , θ ) (M^{2m+1}, \theta ) into a Kähler manifold ( N n , g ) (N^n, g) . We also define and study the notion of pseudo-Hermitian harmonicity for maps from M M into N N . In particular, we prove a CR version of the Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a Kähler manifold having strongly negative curvature.

On the Yamabe problem on contact Riemannian manifolds

A contact Riemannian manifold, whose complex structure is not necessarily integrable, is the generalization of the notion of a pseudohermitian manifold in CR geometry. The Tanaka–Webster–Tanno connection plays the role of the Tanaka–Webster connection for a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By using special frames and normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, the Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. So the Yamabe problem on a contact Riemannian manifold is always solvable.