The contact Yamabe flow on K-contact manifolds

We use the contact Yamabe flow to find solutions of the contact Yamabe problem on K-contact manifolds.     

The contact Yamabe flow on <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We use the contact Yamabe flow to find solutions of the contact Yamabe problem on K-contact manifolds.      

Equivariant Yamabe problem and Hebey-Vaugon conjecture

In their study of the Yamabe problem in the presence of isometry groups, E. Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T. Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey-Vaugon conjecture in dimensions less or equal to 37.     

Pseudo-conformal quaternionic CR structure on (4n + 3)-dimensional manifolds

We study an integrable, nondegenerate codimension 3-subbundle ${\mathcal{D}}We study an integrable, nondegenerate codimension 3-subbundle on a (4n + 3)-manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an -valued 1-form ω locally on a neighborhood U such that and construct the curvature invariant on (M, ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic K?hler 4n-manifolds. From this viewpoint, we exhibit a quaternionic analogue of Chern-Moser’s CR structure. The authors are grateful to ESI for financial support and hospitality during the preparation of this work. The first author acknowledge the support by Grant FWF Project P17108-N04 (Vienna) and Grant N MSM 0021622409 of the Ministry of Education, Youth and Sports (Brno).     

四元Heisenberg群上的链和R-圆及其性质

定义了四元双曲空间上的链和R-圆,并给出了链在垂直投影下的性质.证明了经过Heisenberg群上固定两点的链的唯一性,R-球的qc-水平性,并给出了R-圆与纯虚R-圆之间的关系.     

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.     

Compactness of solutions to the Yamabe problem. III

For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates would imply the compactness of the set of solutions of the Yamabe equation on such manifolds.     

Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the(1 + n)-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n = 2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem;while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.     

Perturbation of Yamabe equation on Iwasawa N groups in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. Under the assumption of some symmetries, we obtain an existent result for the nonlinear equation △NU ＋ （1 ＋ ∈K（x, z））u2＊-1 = 0 on N, which generalizes the result of Malchiodi and Uguzzoni to the Kohn＇s subelliptic context on N in presence of symmetry.     

The second Yamabe invariant

Let (M,g) be a compact Riemannian manifold of dimension n?3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.     

The Yamabe problem for almost Hermitian manifolds

The conformal class of a Hermitian metric g on a compact almost complex manifold (M^2m, J) consists entirely of metrics that are Hermitian with respect to J. For each one of these metrics, we may define a J-twisted version of the Ricci curvature, the J-Ricci curvature, and its corresponding trace, the J-scalar curvature s^J. We ask if the conformal class of g carries a metric with constant s^J, an almost Hermitian version of the usual Yamabe problem posed for the scalar curvature s. We answer our question in the affirmative. In fact, we show that (2m−1)s^J−s=2(2m−1)W(ω, ω), where W is the Weyl tensor and ω is the fundamental form of g. Using techniques developed for the solution of the problem for s, we construct an almost Hermitian Yamabe functional and its corresponding conformal invariant. This invariant is bounded from above by a constant that only depends on the dimension of M, and when it is strictly less than the universal bound, the problem has a solution that minimizes the almost complex Yamabe functional. By the relation above, we see that when W (ω, ω) is negative at least one point, or identically zero, our problem has a solution that minimizes the almost Hermitian Yamabe functional, and the universal bound is reached only in the case of the standard 6-sphere equipped with a suitable almost complex structure. When W(ω, ω) is non-negative and not identically zero, we prove that the conformal invariant is strictly less than the universal bound, thus solving the problem for this type of manifolds as well. We discuss some applications.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

Weak and smooth solutions for a fractional Yamabe flow: The case of general compact and locally conformally flat manifolds

As a counterpart of the classical Yamabe problem, a fractional Yamabe flow has been introduced by Jin and Xiong (2014) on the sphere. Here we pursue its study in the context of general compact smooth manifolds with positive fractional curvature. First, we prove that the flow is locally well posed in the weak sense on any compact manifold. If the manifold is locally conformally flat with positive Yamabe invariant, we also prove that the flow is smooth and converges to a constant fractional curvature metric. We provide different proofs using extension properties introduced by Chang and González (2011) for the conformally covariant fractional order operators.     

From Grushin to Heisenberg via an isoperimetric problem

The Grushin plane is a right quotient of the Heisenberg group. Heisenberg geodesics' projections are solutions of an isoperimetric problem in the Grushin plane.     

四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解,该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解,并且给出了基本解的积分表示.     

Asymptotics for Certain Harmonic Functions and the Martin Compactification on the Quaternionic Heisenberg Group

In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.     

The variational contact problem for elastic objects of different dimensions

We consider the variational free boundary problem describing the contact of an elastic plate with a thin elastic obstacle. The contact domain is unknown a priori and should be determined. The problem is described by a variational inequality for a fourth-order operator. The constraint on the displacement is given on a set of dimension less than that of the solution domain. We find the boundary conditions on the set of the possible contact and their exact statement. We justify the mixed statement of the problem and analyze the limit cases corresponding to the unbounded increase of the elasticity coefficients of the contacting bodies.     

The Yang-Mills functional and Laplace's equation on quantum Heisenberg manifolds

In this paper, we discuss the Yang-Mills functional and a certain family of its critical points on quantum Heisenberg manifolds using noncommutative geometrical methods developed by A. Connes and M. Rieffel. In our main result, we construct a certain family of connections on a projective module over a quantum Heisenberg manifold that gives rise to critical points of the Yang-Mills functional. Moreover, we show that there is a relationship between this particular family of critical points of the Yang-Mills functional and Laplace's equation on multiplication-type, skew-symmetric elements of quantum Heisenberg manifolds; recall that Laplacian is the leading term for the coupled set of equations making up the Yang-Mills equation.