We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant c ⩾ 0 . Our first main result builds on previous results by Yang and establishes of solutions Einstein-Bogomol'nyi equations, corresponding = , in all admissible Kähler classes. second completely solves > Both are proved continuity method require that GIT stability condition an e...

In pointwise differential geometry, i.e., linear algebra, we prove two theorems about the curvature operator of isometrically immersed submanifolds. We restrict our attention to Euclidean immersions because here the results are most easily stated and the curvature operator can be simply expressed as the sum of wedges of second fundamental form matrices. First, we reprove and extend a 1970 resul...

hadamard (or complete $cat(0)$) spaces are complete, non-positive curvature, metric spaces. here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. our results extend the standard non-linear ergodic theorems for non-expansive maps on real hilbert spaces, to non-expansive maps on had...

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...