We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with the (conformal Laplacian power) GJMS operators, a suitable selection of the boundary operators yields formally self-adjoint elliptic conformal boundary problems. Working on a conformal manifold with boundary, we show that the operators yield odd-order conformally invariant fractional Laplacian pseudo-differential operators. To do this, we use higher-order conformally invariant Dirichlet-to-Neumann constructions. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson Q-curvature and include higher-order generalisations of the mean curvature and the T-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We use our constructions to shed light on some conjectures of Juhl.
We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This leads to an explicit description of $Q$-curvature and gauge companion operators on differential forms.
We study the cohomology of the complexes of differential, integral and pseudo forms on odd symplectic manifolds taking the wedge product with the symplectic form as differential. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham dif…
Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.
We study an optimal control problem associated to the conformal Laplacian obstacle problem on closed n-dimensional Riemannian manifolds with n >2. When the Yamabe invariant of the Riemannian manifold is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal metric with constant scalar curvature. For the standard sphere, we prove that the standard bubbles -- namely conformal factor of metrics conformal to the standard one with constant positive scalar curvature -- are the only optimal …
In this paper, we introduce a new functional for the conformal spectrum of the conformal laplacian on a closed manifold M of dimension at least 3. For this new functional we provide a Korevaar type result. The main body of the paper deals with the case of the sphere but a section is devoted to more general closed manifolds.
Let $(X,g)$ be a compact manifold with boundary $M^n$ and $σ$ a defining function of $M$. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric $σ^{-2}g$. We prove that if $σ$ is an approximate solution of a singular Yamabe problem, these families can be wr…
We study the Laplacian flow of a $\mathrm{G}_2$-structure where this latter structure is claimed to be Locally Conformal Parallel. The first examples of long time solutions of this flow with the Locally Conformal Parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. These examples are one-parameter families of Locally Conformal Parallel $\mathrm{G}_2$-structures on rank-one solvable extensions of six-dimensional nilpotent Lie groups. The found solutions are used to construct long time solutions to the Laplacian coflow starting from a Locally Co…
The spectrum of magnetic Schrödinger operators and $k$-form Laplacians on conformally cusp manifolds
我们考虑具有边界的紧致带边流形内部的开流形，并且黎曼度量在边界附近渐近于共形圆柱度量。我们表明，在磁场的存在下，不定义一个完整的相对上同调类的功能的拉普拉斯算子的本质谱消失。由此可见，即使在强磁场的作用下，本质谱也不稳定。我们也处理磁场经营者扰动电场。在同一背景下，我们描述的本质谱的$k$形式的拉普拉斯算子。当边界的$k$和$k-1$ de Rham上同调群消失时，这一点恰好消失。在所有纯点谱情形下，我们给出了特征值计数函数的Weyl型渐近性。在其他情况下，我们描述本质谱。
Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. We study the variational properties of the $k$-th eigenvalue functional $\tilde g\in[g] \mapsto λ_k(L_{\tilde g})$ under a non-volume normalization proposed by Sire-Xu. We discuss necessary conditions for the existence of extremal eigenvalues under such normalization. Also, we discuss the general existence problem when $k=1$.
Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with boundary. In this article, we study the effects of the presence of a nontrivial conformal vector field on $(M^n,g)$. We used the wekk-known de-Rham Laplace operator and a nontrivial solution of the famous Fischer-Marsden differential equation to provide two characterizations of the hemisphere $\mathbb{S}^{n}_{+}(c)$ of constant curvature $c>0.$ As a consequence of the characterization using the Fischer-Marsden equation, we prove the cosmic no-hair conjecture under a given integral condition.
We show that the periodic $η$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between manifolds of dimensions $4$ and $6$. The proof combines a relative version of the Schoen--Yau minimal surface technique with an end-periodic index theorem for the Dirac operator. As a result, we show that the bordism groups $Ω^{spin,+}_{n+1}(S^1 \times BG)$ are infinite for any non-trivial group $G$ which is the fundamental group of a spin spherical space form of dimension $n=3$ or $5$.
