In 1983, Paneitz [23] introduced a conformally fourth order operator defined on 4-dimensional Riemannian manifolds. Branson [1] generalized the definition to n-dimensional Riemannian manifolds, n ≥ 5. Such operators have a geometrical meaning. While the conformal Laplacian is associated to the scalar curvature, the Paneitz-Branson operator is associated to a notion of Q-curvature. Possible refe...

We exhibit a curious link between the Quadratic Orthogonal Bisectional Curvature, combinatorics, and distance geometry. The Weitzenb\"ock curvature operator, acting on real (1,1)--forms, is realized as Dirichlet energy of finite graph, weighted by matrix curvature. These results also illuminate difference in nature Curvature Real Curvature.

In this paper we study the r-stability of closed hypersurfaces with constant r-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the r-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the r-th mean curvature.

Curvature collineations of a spray manifold induced by the Lie symmetries of the underlying spray are studied. The basic observation is that the Jacobi endomorphism and the Berwald curvature are invariant under these symmetries; this implies the invariance of the further curvature data. Our main technical tool is an appropriate Lie derivative operator along the tangent bundle projection. M.S.C....

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclidean space C . We also show that the volume growth condition can be removed if ...

ABSTRACT. We study the generalization of the Biot-Savart law from electrodynamics in the presence of curvature. We define the integral operator BS acting on all vector fields on subdomains of the three-dimensional sphere. By doing so, we establish a geometric setting for electrodynamics in positive curvature. When applied to a vector field, the BiotSavart operator behaves like a magnetic field;...

We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, C3, surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error esti...

A 4-manifold with b+ > 1 and a nonvanishing Seiberg–Witten invariant cannot admit a metric of positive scalar curvature. This remarkable fact is proved [18] using the Weitzenböck–Lichnerowicz formula for the square of the Spin Dirac operator, combined with the ‘curvature’ part of the Seiberg–Witten equations. Thus, in dimension 4, there is a strong generalization of Lichnerowicz’s vanishing the...

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy th...

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichmüller space Teich(Sg) of a closed surface Sg of genus g is uniformly bounded away...