Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to f...Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows: $$ {\mathcal{P}_\alpha} = {\mathcal{L}_\alpha} \bar {\mathcal{L}_\alpha} = \frac{1} {4}\left[ {\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} } \right]^2 + \alpha ^2 T^2 $$ Here “Z j ” j=1 n is an orthonormal basis for the subbundle T (1,0) of the complex tangent bundle T ?(H n ) and T is the “missing direction”. The operator $ \mathcal{L}_\alpha $ is the sub-Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λ α . We also construct projection operators and relative fundamental solution for the operator $ \mathcal{L}_\alpha $ while α ∈ Λ α .展开更多
CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformall...CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three. While the sign of this operator is important in the embedding problem, the kernel of this operator is also closely connected with the stability of CR structures. The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed. The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem. The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension, and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms. Finally, an isoperimetric constant determined by the Q-prime curvature integral is discussed.展开更多
In this paper,we first establish the existence of blow-up solutions with two antipodal points of the fourth order mean field equations on S^(4).Moreover,we construct non-axially symmetric solutions with blow-up points...In this paper,we first establish the existence of blow-up solutions with two antipodal points of the fourth order mean field equations on S^(4).Moreover,we construct non-axially symmetric solutions with blow-up points at the vertices of regular configurations,i.e.,equilateral triangles on a great circle,regular tetrahedrons,cubes,octahedrons,icosahedrons and dodecahedrons.The bubbling rates of these blow-up solutions rely on various bubbling configurations.展开更多
基金supported by a research grant from the United States Air Force Office of Scientific Research(AFOSR) SBIR Phase I (Grant No. FA9550-09-C-0045)a Hong Kong RGC competitive earmarked research(Grant No. 600607)+1 种基金a competitive research grant at Georgetown University (Grant No. GD2236000)supported by Natural Science Foundation of Taiwan,China (Grant No.97-2115-M-002-015)
文摘Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows: $$ {\mathcal{P}_\alpha} = {\mathcal{L}_\alpha} \bar {\mathcal{L}_\alpha} = \frac{1} {4}\left[ {\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} } \right]^2 + \alpha ^2 T^2 $$ Here “Z j ” j=1 n is an orthonormal basis for the subbundle T (1,0) of the complex tangent bundle T ?(H n ) and T is the “missing direction”. The operator $ \mathcal{L}_\alpha $ is the sub-Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λ α . We also construct projection operators and relative fundamental solution for the operator $ \mathcal{L}_\alpha $ while α ∈ Λ α .
文摘CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three. While the sign of this operator is important in the embedding problem, the kernel of this operator is also closely connected with the stability of CR structures. The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed. The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem. The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension, and it is closely connected with the pluriharmonic functions.The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms. Finally, an isoperimetric constant determined by the Q-prime curvature integral is discussed.
基金supported by National Science Foundation of USA (Grant No.DMS-1901914)supported by National Natural Science Foundation of China (Grant Nos.12101612 and 12171456)。
文摘In this paper,we first establish the existence of blow-up solutions with two antipodal points of the fourth order mean field equations on S^(4).Moreover,we construct non-axially symmetric solutions with blow-up points at the vertices of regular configurations,i.e.,equilateral triangles on a great circle,regular tetrahedrons,cubes,octahedrons,icosahedrons and dodecahedrons.The bubbling rates of these blow-up solutions rely on various bubbling configurations.
