Renato G. Bettiol

Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many ts accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.

Delaunay-Type Hypersurfaces in Cohomogeneity One Manifolds

Classical Delaunay surfaces are highly symmetric constant mean curvature (CMC) submanifolds of space forms. We prove the existence of Delaunay-type hypersurfaces in a large class of compact manifolds, using the geometry of cohomogeneity one group actions and variational bifurcation techniques. Our construction specializes to the classical examples in round spheres, and allows to obtain Delaunay-type hypersurfaces in many other ambient spaces, ranging from complex and quaternionic projective spaces to Kervaire exotic spheres.

Positive biorthogonal curvature on S 2 ×S 2

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, is strictly positive; and this can be done with an arbitrarily small lower bound on the distance between the planes considered. Although they have positive Ricci curvature, these metrics do not have nonnegative sectional curvature. Such metrics also have positive biorthogonal curvature, meaning that the average of sectional curvatures of any two orthogonal planes is positive.

Three-manifolds with many flat planes

We discuss the rigidity (or lack thereof) imposed by dierent no- tions of having an abundance of zero curvature planes on a complete Riemann- ian 3-manifold. We prove a rank rigidity theorem for complete 3-manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study 3-manifolds such that every tangent vector is con- tained in a at plane, including examples with irreducible universal covering, and discuss the eect of nite volume and real-analiticity assumptions.

On the Equivariant Implicit Function Theorem with Low Regularity and Applications to Geometric Variational Problems

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-)Riemannian geodesics, and constant mean curvature hypersurfaces.

Strongly nonnegative curvature

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

Flag manifolds with strongly positive curvature

We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds

Positive biorthogonal curvature on

W^6