arXiv subjects
Publications and source records attributed to Paul Yang.
We study a CR-invariant equation for vanishing $E_1$ surfaces in the 3-dimensional Heisenberg group. This is shown to be a hyperbolic equation. We prove the local uniqueness theorem for an initial value problem and classify all such global surfaces with rotational symmetry. We also show that the Clifford torus in the CR 3-sphere is not a local minimizer of $E_1$ by computing the second variation.
In this paper, we prove several rigidity and quantitative rigidity results for asymptotically hyperbolic Poincar\'e-Einstein manifolds whose conformal infinities are diffeomorphic to a cylinder $S^1 \times S^{n - 1}$. It is a basic fact that the Riemannian product $S^1 \times S^{n - 1}$ can bound, in addition to a complete hyperbolic metric on $S^1 \times D^n$, other Poincar\'e-Einstein metrics such as the AdS-Schwarzschild metrics on $D^2 \times S^{n - 1}$. The main result shows that any Poincar\'e-Einstein filling of $S^1 \times S^{n - 1}$ must be hyperbolic if it is non-positively curved. As corollaries, the Poincar\'e-Einstein filling of $S^1 \times S^{n - 1}$ is unique when the length of circle factor is sufficiently large or the $L^2$-energy of the Weyl curvature is sufficiently small relative to the Yamabe constant of the conformal infinity. To prove the Weyl pinching rigidity, we established a new $\epsilon$-regularity for the Weyl curvature of a general class of Poincar\'e-Einstein manifolds with conformal infinity of positive Yamabe type, which includes non-compact and volume-collapsed families of Poincar\'e-Einstein spaces in all dimensions.
We propose a novel energy-aware federated learning (FL)-based system, namely SusFL, for sustainable smart farming to address the challenge of inconsistent health monitoring due to fluctuating energy levels of solar sensors. This system equips animals, such as cattle, with solar sensors with computational capabilities, including Raspberry Pis, to train a local deep-learning model on health data. These sensors periodically update Long Range (LoRa) gateways, forming a wireless sensor network (WSN) to detect diseases like mastitis. Our proposed SusFL system incorporates mechanism design, a game theory concept, for intelligent client selection to optimize monitoring quality while minimizing energy use. This strategy ensures the system's sustainability and resilience against adversarial attacks, including data poisoning and privacy threats, that could disrupt FL operations. Through extensive comparative analysis using real-time datasets, we demonstrate that our FL-based monitoring system significantly outperforms existing methods in prediction accuracy, operational efficiency, system reliability (i.e., mean time between failures or MTBF), and social welfare maximization by the mechanism designer. Our findings validate the superiority of our system for effective and sustainable animal health monitoring in smart farms. The experimental results show that SusFL significantly improves system performance, including a $10\%$ reduction in energy consumption, a $15\%$ increase in social welfare, and a $34\%$ rise in Mean Time Between Failures (MTBF), alongside a marginal increase in the global model's prediction accuracy.
We show that the half-ball in $\mathbb{R}^4$ can be conformally changed so that the only contribution to the Gauss--Bonnet formula is a constant term at the corner. This may be seen as a fourth-order Cherrier--Escobar-type problem on the half-ball.
We characterize homogeneous three-dimensional CR manifolds, in particular Rossi spheres, as critical points of a certain energy functional that depends on the Webster curvature and torsion of the pseudohermitian structure.
In this paper we compute the first and second variation of the normalized Einstein-Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater or equal to five, in three dimension we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.
We discuss the solution of the Neumann problem associated with the CR Yamabe operator on a subset $Ω$ of the CR manifold $\mathbb{S}^3$ bounded by the Clifford torus $Σ$. We also discuss the Yamabe-type problem of finding a contact form on $Ω$ which has zero Tanaka--Webster scalar curvature and for which $Σ$ has constant $p$-mean curvature.
We prove that every closed, universally embeddable CR three-manifold with nonnegative Yamabe constant and positive total $Q^\prime$-curvature is contact diffeomorphic to a quotient of the standard contact three-sphere. We also prove that every closed, embeddable CR three-manifold with zero Yamabe constant and nonnegative total $Q^\prime$-curvature is CR equivalent to a compact quotient of the Heisenberg group with its flat CR structure.
We apply scattering theory on asymptotically hyperbolic manifolds to singular Yamabe metrics, applying the results to the study of the conformal geometry of compact manifolds with boundary. In particular, we define extrinsic versions of the conformally invariant powers of the Laplacian, or GJMS operators, on the boundary of any such manifold, along with associated extrinsic Q-curvatures. We use the existence and uniqueness of a singular Yamabe metric to define also nonlocal extrinsic fractional GJMS operators on the boundary, and draw other global conclusions about the scattering operator, including a Gauss-Bonnet theorem in dimension four.
We study integral curvature conditions for a Riemannian metric $g$ on $S^4$ that quantify the best bilipschitz constant between $(S^4,g)$ and the standard metric on $S^4$. Our results show that the best bilipschitz constant is controlled by the $L^2$-norm of the Weyl tensor and the $L^1$-norm of the $Q$-curvature, under the conditions that those quantities are sufficiently small, $g$ has a positive Yamabe constant and the $Q$-curvature is mean-positive. The proof of the result is achieved in two steps. Firstly, we construct a quasiconformal map between two conformally related metrics in a positive Yamabe class. Secondly, we apply the Ricci flow to establish the bilipschitz equivalence from such a conformal class to the standard conformal class on $S^4$.
We prove rigidity for the Lichnerowicz-type eigenvalue estimate for the Kohn Laplacian on strictly pseudoconvex three-manifolds with nonnegative CR Paneitz operator and positive Webster curvature.
We exhibit examples of compact three-dimensional CR manifolds of positive Webster class, {\em Rossi spheres}, for which the pseudo-hermitian mass as defined in \cite{CMY17} is negative, and for which the infimum of the CR-Sobolev quotient is not attained. To our knowledge, this is the first geometric context on smooth closed manifolds where this phenomenon arises, in striking contrast to the Riemannian case.
The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work \cite{WY}, we have proved that if the $Q'$-curvature is nonnegative, and the integral of $Q'$-curvature is below the dimensional bound $c_1'$, then we have the isoperimetric inequality. In this paper, we manage to drop the condition on the nonnegativity of the $Q'$-curvature. We prove that the volume form $e^{4u}$ is a strong $A_\infty$ weight. As a corollary, we prove the Sobolev-Poincaré inequality on a class of CR-manifolds with integrable $Q'$-curvature.
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 the coefficient (up to a constant multiple) of the log term in the associated volume renormalization.
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.
In this paper, we study contact forms on the three- dimensional Heisenberg manifold with its standard CR structure. We discover that the $Q'$-curvature, introduced by Branson, Fontana and Morpurgo [BFM13] on the CR three-sphere and then generalized to any pseudo-Einstein CR three manifold by Case and Yang [CY95], controls the isoperimetric inequality on such a CR-manifold. To show this, we first prove that the nonnegative Webster curvature at infinity deduces that the metric is normal, which is analogous to the behavior on a Riemannian four-manifold.
We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher dimensions for two cases: (a) the touching point is nonsingular; (b) the touching point is an isolated singular point for one of comparison functions. For a background subriemannian manifold with local symmetry of isometric translations, we have the strong maximum principle for associated graphs which include, among others, intrinsic graphs with constant horizontal (p-) mean curvature. As applications, we show a rigidity result of horizontal (p-) minimal hypersurfaces in any higher dimensional Heisenberg cylinder and a pseudo-halfspace theorem for any Heisenberg group.
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 $\sqrt{P^\prime}$.