37 citations · 244 across the 33 of their papers we have counts for
35 papers
Counting Homogeneous Einstein Metrics
Renato G. Bettiol, Hannah Friedman
We present an explicit upper bound on the number of isolated homogeneous Einstein metrics on compact homogeneous spaces whose isotropy representations consist of pairwise inequival…
Bifurcations of Clifford tori in ellipsoids
Renato G. Bettiol, Paolo Piccione
We prove that 3-dimensional ellipsoids invariant under a 2-torus action contain infinitely many distinct immersed minimal tori, with at most one exception. These minimal tori bifur…
Multiplicity of singular solutions to the fractional Yamabe problem on spheres
Renato G. Bettiol, María del Mar González, Ali Maalaoui
We prove nonuniqueness results for complete metrics with constant positive fractional curvature conformal to the round metric on , using bifurcation techniques.…
Curvature operators and rational cobordism
Renato G. Bettiol, McFeely Jackson Goodman
We determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted genus on a closed Riemannian spin manifold, where the twisti…
Diameter and displacement of sphere involutions
Renato G. Bettiol, Emilio A. Lauret
We show that spheres in all dimensions can be deformed to have diameter larger than the distance between any pair of antipodal points. This answers a question of Yurii Niko…
Extremality and rigidity for scalar curvature in dimension four
Renato G. Bettiol, McFeely Jackson Goodman
Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that la…
Ricci flow does not preserve positive sectional curvature in dimension four
Renato G. Bettiol, Anusha M. Krishnan
We find examples of cohomogeneity one metrics on and with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are…
Nonplanar minimal spheres in ellipsoids of revolution
Renato G. Bettiol, Paolo Piccione
We use global bifurcation techniques to establish the existence of arbitrarily many geometrically distinct nonplanar embedded smooth minimal 2-spheres in sufficiently elongated 3-d…
Global bifurcation for a class of nonlinear ODEs
Renato G. Bettiol, Paolo Piccione
We briefly survey global bifurcation techniques, and illustrate their use by finding multiple positive periodic solutions to a class of second order quasilinear ODEs related to the…
Geography of pinched four-manifolds
Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes
We prove several new restrictions on the Euler characteristic and signature of oriented 4-manifolds with (positively or negatively) pinched sectional curvature. In particular, we s…
The first eigenvalue of a homogeneous CROSS
Renato G. Bettiol, Emilio A. Lauret, Paolo Piccione
We provide explicit formulae for the first eigenvalue of the Laplace-Beltrami operator on a compact rank one symmetric space (CROSS) endowed with any homogeneous metric. As consequ…
Convex Algebraic Geometry of Curvature Operators
Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes
We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More p…
Nonuniqueness of conformal metrics with constant -curvature
Renato G. Bettiol, Paolo Piccione, Yannick Sire
We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) -curvature on compact and noncompact manifolds of d…
Sectional curvature and Weitzenböck formulae
Renato G. Bettiol, Ricardo A. E. Mendes
We establish a new algebraic characterization of sectional curvature bounds and using only curvature terms in the Weitzenböck formulae for symmetric -t…
Teichmüller theory and collapse of flat manifolds
Renato G. Bettiol, Andrzej Derdzinski, Paolo Piccione
We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry c…
Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature
Renato G. Bettiol, Anusha M. Krishnan
We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.
Infinitely many solutions to the Yamabe problem on noncompact manifolds
Renato G. Bettiol, Paolo Piccione
We establish the existence of infinitely many complete metrics with constant scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include…
Strongly nonnegative curvature
Renato G. Bettiol, Ricardo A. E. Mendes
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…
Four-dimensional manifolds with positive biorthogonal curvature
Renato G. Bettiol
We classify, up to homeomorphisms, the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes a…
Flag manifolds with strongly positive curvature
Renato G. Bettiol, Ricardo A. E. Mendes
We obtain a complete description of the moduli spaces of homogeneous metrics with strongly positive curvature on the Wallach flag manifolds , and , which are…
Deformations of free boundary CMC hypersurfaces
Renato G. Bettiol, Paolo Piccione, Bianca Santoro
We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean…
Three-manifolds with many flat planes
Renato G. Bettiol, Benjamin Schmidt
We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian 3-manifold. We prove a rank rigidity…
Deforming solutions of geometric variational problems with varying symmetry groups
Renato G. Bettiol, Paolo Piccione, Gaetano Siciliano
We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by…
Strongly positive curvature
Renato G. Bettiol, Ricardo A. E. Mendes
We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and ste…