The $Q$-prime curvature is a local pseudo-Einstein invariant on CR manifolds defined by Case and Yang, and Hirachi. Its integral, the total $Q$-prime curvature, gives a non-trivial global CR invariant. On the other hand, Marugame has constructed a family of global CR invariants via renormalized characteristic forms, which contains the total $Q$-prime curvature. In this paper, we introduce a generalization of the $Q$-prime curvature for each renormalized characteristic form, and show that its integral coincides with Marugame's CR invariant. We also study generalizations of the critical CR GJMS operator and the $P$-prime operator, which are related to the transformation laws of our new curvatures under conformal change.
We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient $v_3$ in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are stable with respect to the associated $\mathcal{W}$-functional.
For each invariant polynomial $Φ$, we construct a global CR invariant via the renormalized characteristic form of the Cheng--Yau metric on a strictly pseudoconvex domain. When the degree of $Φ$ is 0, the invariant agrees with the total $Q'$-curvature. When the degree is equal to the CR dimension, we construct a primed pseudo-hermitian invariant $\mathcal{I}'_Φ$ which integrates to the corresponding CR invariant. These are generalizations of the $\mathcal{I}'$-curvature on CR five-manifolds, introduced by Case--Gover.
We establish an algorithm which computes formulae for the CR GJMS operators, the $P^\prime$-operator, and the $Q^\prime$-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the $P^\prime$-operator, and shows that the $Q^\prime$-curvature is constant, with the constant explicitly given in terms of the Webster scalar curvature. We also use our algorithm to derive local formulae for the $P^\prime$-operator and $Q^\prime$-curvature of a five-dimensional pseudo-Einstein manifo…
We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.
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.
Computing renormalized curvature integrals on Poincaré-Einstein manifolds
Penn State University（宾夕法尼亚州立大学）；Rutgers University（罗格斯大学）；Wichita State University（威奇托州立大学）；University of Notre Dame（圣母大学）；Sun Yat-sen University（中山大学）
We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gover and the first author, we study the property of CR invariant differential operator of order 2(n+3) on 2n+1 dimensional CR manifolds that appears in the second variational formula of the total Q-prime curvature.
We construct higher-dimensional analogues of the $\mathcal{I}^\prime$-curvature of Case and Gover in all CR dimensions $n\geq2$. Our $\mathcal{I}^\prime$-curvatures all transform by a first-order linear differential operator under a change of contact form and their total integrals are independent of the choice of pseudo-Einstein contact form on a closed CR manifold. We exhibit examples where these total integrals depend on the choice of general contact form, and thereby produce counterexamples to the Hirachi conjecture in all CR dimensions $n\geq2$.
The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. In this paper, we generalize these objects to the boundaries of asymptotically complex hyperbolic Einstein (ACHE) manifolds, which are partially integrable, strictly pseudoconvex CR manifolds, by using the scattering matrix for ACHE manifolds. In this setting, the $P$-prime opera…
We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.
We construct contact forms with constant $Q^\prime$-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the $II$-functional from conformal geometry. Two crucial steps are to show that the $P^\prime$-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green's function for $\sqrt{P^\prime}$.
We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For th…
The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family $\H(r;g)$ of self-adjoint elliptic differential operators. $\H(r;g)$ is a non-Laplace-type perturbation of the conformal Laplacian $P_2(g) = \H(0;g)$. It is defined in terms of the metric $g$ and covariant derivatives of the curvature of $g$. We study the heat kernel coefficients $a_{2k}(r;g)$ of $\H(r;g)$ on closed manifolds. We prove general structural results for the heat kernel…
The theory of ambient spaces is useful to define CR invariant objects, such as CR invariant powers of the sub-Laplacian, the $P$-prime operators, and $Q$-prime curvature. However in general, it is difficult to write down these objects in terms of the Tanaka-Webster connection. In this paper, we give those explicit formulas for CR manifolds satisfying an Einstein condition, called Sasakian $η$-Einstein manifolds. As an application, we study properties of the first and the second variation of the total $Q$-prime curvature at Sasakian $η$-Einstein manifolds.
In this paper we study the problem of prescribing the $\bar Q^{\prime}$-curvature on pseudo-Einstein CR 3-manifolds. In the first stage we study the problem in the compact setting and we show that under natural assumptions, one can prescribe any positive CR pluriharmonic function. In the second stage we study the problem in the non-compact setting of the Heisenberg group. Under mild assumptions on the prescribed function, we prove the existence of a one parameter family of solutions. In fact, we show that one can find two kinds of solutions: normal ones that satisfy an isoperimetric inequality…
In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of $\hat{G}=G\times T$ on the free loop space $λM$ of $M$. In particular, for each closed geodesic, we have a $\hat{G}$-orbit of closed geodesics. We assume the Finsler sphere $(M,F)$ described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the …
We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula $$ \mathrm{Ric}(γ^{\prime}\left( 0\right) ,γ^{\prime}\left( 0\right) )=\frac{1}{2}\frac{d^{2}}{ds^{2}}\mathrm{Ric}_{Δ_g}(x,γ\left( s\right) )$$ for any curve $γ(s).$
Local and global conformal invariants of submanifolds
Penn State University（宾夕法尼亚州立大学）；Rutgers University（罗格斯大学）；Wichita State University（威奇托州立大学）；University of Notre Dame（圣母大学）；Sun Yat-sen University（中山大学）
The nonnegativity of the CR Paneitz operator plays a crucial role in three-dimensional CR geometry. In this paper, we prove this nonnegativity for embeddable CR manifolds. This result and previous works give an affirmative solution of the CR Yamabe problem for embeddable CR manifolds. We also show the existence of a contact form with zero CR $Q$-curvature, and generalize the total $Q$-prime curvature to embeddable CR manifolds with no pseudo-Einstein contact forms. Furthermore, we discuss the logarithmic singularity of the Szegő kernel.
