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A Monge-Ampère equation in conformal geometry

Matthew J. Gursky (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider the Monge-Ampère-type equation det ( A + λ g ) = const . , where A is the Schouten tensor of a conformally related metric and λ > 0 is a suitably chosen constant. When the scalar curvature is non-positive we give necessary and sufficient conditions for the existence of solutions. When the scalar curvature is positive and the first Betti number of the manifold is non-zero we also establish existence. Moreover, by adapting a construction of Schoen, we show that solutions are in general not unique.

A note on conformal vector fields on a Riemannian manifold

Sharief Deshmukh, Falleh Al-Solamy (2014)

Colloquium Mathematicae

We consider an n-dimensional compact Riemannian manifold (M,g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ > 0, together with an upper bound on the energy of the vector field ξ, implies that M is isometric to the n-sphere Sⁿ(λ). We also introduce the notion of φ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres...

A Singularity Theorem for Twistor Spinors

Florin Alexandru Belgun, Nicolas Ginoux, Hans-Bert Rademacher (2007)

Annales de l’institut Fourier

We study spin structures on orbifolds. In particular, we show that if the singular set has codimension greater than 2, an orbifold is spin if and only if its smooth part is. On compact orbifolds, we show that any non-trivial twistor spinor admits at most one zero which is singular unless the orbifold is conformally equivalent to a round sphere. We show the sharpness of our results through examples.

A strong maximum principle for the Paneitz operator and a non-local flow for the Q -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

In this paper we consider Riemannian manifolds ( M n , g ) of dimension n 5 , with semi-positive Q -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive Q -curvature. Modifying the test function construction of Esposito-Robert, we show...

Application of spaces of subspheres to conformal invariants of curves and canal surfaces

Rémi Langevin, Jun O'Hara, Shigehiro Sakata (2013)

Annales Polonici Mathematici

We review some techniques from the Möbius geometry of curves and surfaces in the 3-sphere, consider canal surfaces using their characteristic circles, and express the conformal curvature, and conformal torsion, of a vertex-free space curve in terms of its corresponding curve of osculating circles, and osculating spheres, respectively. We accomplish all of this strictly within the framework of Möbius geometry, and compare our results with the literature. Finally, we show how our formulation allows...

Commuting linear operators and algebraic decompositions

Rod A. Gover, Josef Šilhan (2007)

Archivum Mathematicum

For commuting linear operators P 0 , P 1 , , P we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition P = P 0 P 1 P in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem P u = f reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential...

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