We transform the problem of determining isometric immersions from into into that of solving equations of degenerate Monge-Ampère type on the unit ball . By presenting one family of special solutions to the equations, we obtain a great many noncongruent examples of such isometric immersions with or without umbilic set.
We provide a classification of isometries of systolic complexes corresponding to the classification of isometries of CAT(0)-spaces. We prove that any isometry of a systolic complex either fixes the barycentre of some simplex (elliptic case) or stabilizes a thick geodesic (hyperbolic case). This leads to an alternative proof of the fact that finitely generated abelian subgroups of systolic groups are undistorted.
We discuss the notion of isotropic curvature of a Riemannian manifold and relations between the sign of this curvature and the geometry and topology of the manifold.
We present some examples of Killing tensors and give their geometric interpretation. We give new examples of non-compact complete and compact Riemannian manifolds whose Ricci tensor ϱ satisfies the condition
For a given U(1)-bundle E over M = {x1, ..., xn}, where the xi are n distinct points of , we minimise the U(1)-Higgs action and we make an asymptotic analysis of the minimizers when the coupling constant tends to infinity. We prove that the curvature (= magnetic field) converges to a limiting curvature that we give explicitely and which is singular along line vortices which connect the xi. This work is the three dimensional equivalent of previous works in dimension two (see [3] and [4]). The...
In the first part of this paper, we prove local interior and boundary gradient estimates for -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the -harmonic...