Infinitely many solutions to the Yamabe problem on noncompact manifolds

TL;DR: The existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain non-compact product manifolds has been established in this article, including products of closed manifolds and simply-connected symmetric spaces of Euclidean type.

Abstract: We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $\mathbb S^m \times\mathbb R^d$, $m\geq2$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d

TL;DR: The first eigenvalue of the Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric is given in this article.

Abstract: We provide explicit formulae for the first eigenvalue of the Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequences, we prove that homogeneous metrics on CROSSes are isospectral if and only if they are isometric, and also discuss their stability (or lack thereof) as solutions to the Yamabe problem.

TL;DR: In this paper, the authors investigated special spinorial Yamabe metrics on product manifolds and developed a bubbling analysis which has independent interest in the present setting, motivated to bubbling phenomena for the Riemannian problem and recent multiplicity results in this setting.

Abstract: This work is devoted to the analysis of the Yamabe problem on Spin manifolds and some applications to CMC immersions. Despite the efforts of many authors, very little is known on the existence of Yamabe metrics on general Spin manifolds. Motivated to bubbling phenomena for the Riemannian problem and recent multiplicity results in this setting, we investigate special spinorial Yamabe metrics on product manifolds developing a bubbling analysis which has independent interest in the present setting.

TL;DR: In this article, the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and non-compact manifolds of dimension $geq5 was studied.

Abstract: We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifurcate from Berger metrics on spheres and complex projective spaces. These provide examples of nonisometric metrics with the same constant negative $Q$-curvature in a conformal class with negative Yamabe invariant, echoing the absence of a Maximum Principle. We also discover infinitely many complete metrics with constant $Q$-curvature conformal to $\mathbb S^m\times\mathbb R^d$, $m\geq4$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d\leq m-3$; which give infinitely many solutions to the singular constant $Q$-curvature problem on round spheres $\mathbb S^n$ blowing up along a round subsphere $\mathbb S^k$, for all $0\leq k<(n-4)/2$.

TL;DR: In this paper, the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and non-compact manifolds of dimension $geq5 was studied.

Abstract: We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifurcate from Berger metrics on spheres and complex projective spaces. These provide examples of nonisometric metrics with the same constant negative $Q$-curvature in a conformal class with negative Yamabe invariant, echoing the absence of a Maximum Principle. We also discover infinitely many complete metrics with constant $Q$-curvature conformal to $\mathbb S^m\times\mathbb R^d$, $m\geq4$, $d\geq1$, and $\mathbb S^m\times\mathbb H^d$, $2\leq d\leq m-3$; which give infinitely many solutions to the singular constant $Q$-curvature problem on round spheres $\mathbb S^n$ blowing up along a round subsphere $\mathbb S^k$, for all $0\leq k<(n-4)/2$.

TL;DR: A recent workshop on the analysis, geometry and topology of positive scalar curvature metrics was organized by Bernd Ammann (Regensburg), Bernhard Hanke (Augsburg), and André Neves (Chicago).

Abstract: Riemannian manifolds with positive scalar curvature play an important role in mathematics and general relativity. Obstruction and existence results are connected to index theory, bordism theory and homotopy theory, using methods from partial differential equations and functional analysis. The workshop led to a lively interaction between mathematicians working in these areas. Mathematics Subject Classification (2010): Primary: 53C20 (Global Riemannian Geometry). Secondary: 19K56, 35Q75, 53C15, 53C21, 53C25, 53C27, 53C29, 53C42, 53C44, 57R65. Introduction by the Organisers The workshop Analysis, Geometry and Topology of Positive Scalar Curvature Metrics, organised by Bernd Ammann (Regensburg), Bernhard Hanke (Augsburg), and André Neves (Chicago) was attended by more than 50 participants from Europe, the US, and Japan, including a number of young scientists on a doctoral or postdoctoral level. Rather than representing a single mathematical discipline the workshop aimed at bringing together researchers from different areas, but working on similar questions. The conference created a stimulating environment for exchange of ideas and methods from topology, from Riemannian and Lorentzian geometry, and from general relativity. The workshop started with three extended 80 minutes talks by Claude LeBrun, Greg Galloway, and Thomas Schick, introducing to major themes related to the positive scalar curvature problem and appearing again in later talks of the workshop: The notion of mass in Kähler geometry, the role of scalar curvature in 2224 Oberwolfach Report 36/2017 General Relativity, and the application of index theory and differential topology to the classification of positive scalar curvature metrics. Aspects of General Relativity appeared in research talks dealing with the equality case of the spacetime positive mass theorem, topological investigations and new constructions related to horizon geometry, boundary value problems for the static vacuum equations, and investigations of Lorentzian manifolds in terms of Cauchy problems and boundary value problems for the Dirac operator. Recent major results in the subject deal with the topology of spaces and moduli spaces of positive scalar curvature metrics, in particular in connection with our now improved understanding of the diffeomorphism groups of smooth manifolds by the use of cobordism categories. In addition, classical methods, such as the Gromoll filtration combined with the use of Toda brackets, remain vital to obtain interesting new results in this direction. Secondary coarse index theory allows a distinction of concordance classes of uniformly positively curved metrics on non-compact manifolds. Further refinements of index theory were shown to be successful to study the positive scalar curvature problem on Thom-Mather stratified and on non-compact smooth manifolds with controlled geometry at infinity. The positive scalar curvature problem is closely tied to the computation of Yamabe invariants. Some of the talks dealt with the non-uniqueness of solutions to the Yamabe problem on compact and non-compact manifolds, the study of Yamabe invariants on stratified spaces, and the computation of Yamabe invariants by the use of edge-cone Einstein metrics. An important topic of interest related to positive scalar curvature is the geometry of minimal hypersurfaces, which was the subject of talks concerning minimal hypersurfaces with bounded Morse index and the existence of infinitely many geodesics on asymptotically conical surfaces of non-negative scalar curvature. The positive scalar problem also has strong connections to various other parts of Riemannian geometry and global analysis. Stability under Ricci flow of Ricci-flat asymptotically locally Euclidean manifolds ties links between Kähler geometry, scalar curvature geometry, geometric partial differential equations and general relativity. Conformal geometry enters the picture when classifying positive scalar curvature metrics on the seven sphere that cannot be the conformal infinity of Poincaré-Einstein metrics on the eight dimensional ball. A talk about the role of holomorphic sectional curvature in Kähler geometry pointed out an instance when strong curvature assumptions lead to very restrictive classification results. However, the common expectation that the use of stronger curvature notions such as Ricci or sectional curvature always go hand in hand with more restrictive classification results was questioned in three more talks. They dealt with rigidity results in scalar curvature geometry on the one hand, and the application of differential topological methods for a study of moduli spaces of positive Ricci curvature metrics on spheres, and for the construction of multiparameter families of highly connected seven dimensional manifolds admitting metrics of non-negative sectional curvature, on the other. Analysis, Geometry and Topology of Positive Scalar Curvature Metrics 2225 Most research talks had a length of 60 minutes with some additional 40 minutes talks contributed by younger participants. It was interesting to see how researchers originating from distinct mathematical communities dealt with similar problems, but referred to different techniques and sometimes arrived at varying views of the same mathematical structures. Once more the positive scalar curvature problem featured itself as an ideal point of reference to take advantage of an exchange of ideas, tools and perspectives for a productive scientific discussion. Due to the interdisciplinary character of the meeting speakers were asked to keep their lectures at a level accessible to a broad audience with different mathematical backgrounds. A perfect organization and management by the staff of the Oberwolfach institute created an optimal working environment. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1641185, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Dan Lee and Nathan Perlmutter in the “Simons Visiting Professors” program at the MFO. Analysis, Geometry and Topology of Positive Scalar Curvature Metrics 2227 Workshop: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics