In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the $sigma_{2}$-curvature functional.
We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a vir
tual immersion with skew symmetric second fundamental form, if and only if it is a symmetric space, and in this case the virtual immersion is essentially unique.
We exploit an ansatz in order to construct power series expansions for pairs of conjugate functions defined on domains of Euclidean $3$--space. Convergence properties of the resulting series are investigated. Entire solutions which are not harmonic a
re found as well as a $2$-parameter family of examples which contains the Hopf map.
We consider the Ricci flow $frac{partial}{partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rmgeq 0, |Rm(p)|to 0, ~as ~d(o,p)to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.
Positively curved Alexandrov spaces of dimension 4 with an isometric circle action are classified up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always
satisfied. As a corollary, positively curved Riemannian orbifolds of dimension 4 with an isometric circle action are also classified.
We give the first examples of collapsing Ricci limit spaces on which the Hausdorff dimension of the singular set exceeds that of the regular set; moreover, the Hausdorff dimension of these spaces can be non-integers. This answers a question of Cheeger-Colding about collapsing Ricci limit spaces.