arXiv subjects
Publications and source records attributed to Jason DeVito.
We prove that the Ricci flow preserves positive sectional curvature on homogeneous spheres and complex projective spaces. In conjunction with prior results, this completes the classification of which homogeneous spaces have positively curved metrics flowing outside the set of positively curved metrics and which do not.
We develop techniques for solving the relative isoperimetric problem on polygonal domains in $\mathbb{R}^2$, with special attention paid to corners. As an application, we solve the relative isoperimetric problem for a square with a square corner removed.
In each dimension of the form $4n-1$ with $n\geq 3$, we construct infinitely many new examples of manifolds admitting metrics with positive sectional curvature almost everywhere. In addition, we show that if $n\geq 6$, infinitely many of our examples are not homotopy equivalent to any homogeneous space, providing the first infinite family of such examples.
We study the sectional curvature of all of the cohomogeneity two Bazaikin spaces with respect to a Riemannian metric construction due to Wilking. We show that, in contrast to the cohomogeneity one and homogeneous case, for all of the cohomogeneity two examples, the set of points with strictly positive curvature does not have full measure.
We construct metrics of positive $2^{\rm nd}$ intermediate Ricci curvature, $\mathrm{Ric}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including $\mathbb{S}^6\times\mathbb{S}^7$, $\mathbb{S}^7\times\mathbb{S}^7$ and all their simply connected isometric quotients. In particular, we obtain infinitely many examples in dimension 13. We also produce infinitely many non-simply connected spaces with $\mathrm{Ric}_2>0$ in dimensions 13 and 14, including $\mathbb{RP}^6\times \mathbb{RP}^7$ and $\mathbb{RP}^7\times \mathbb{RP}^7$, which cannot admit a metric of positive sectional curvature. The main new idea is a generalization of the concept of fatness which ensures the existence of $\mathrm{Ric}_2>0$ metrics on the total space of certain homogeneous bundles.
We investigate the curvature of Eschenburg spaces with respect to two different metrics, one constructed by Eschenburg and the other by Wilking. With respect to the Eschenburg metric, we obtain a simple complete characterization of the curvature of every Eschenburg space in terms of the triples of integers defining the space. With respect to Wilking's metric, we study all the examples whose natural isometry group acts with cohomogeneity two. Here, we find that apart from the previously known examples with almost positive curvature, all the remaining examples have open sets of points with zero-curvature planes.
A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The Double Soul Conjecture asserts that a closed simply connected manifold admitting a metric of non-negative sectional curvature is necessarily a double disk bundle. We study a generalization of this conjecture by dropping the requirement that the manifold be simply connected. Previously, a unique counterexample was known to this generalization, the Poincar\'e dodecahedral space $S^3/I^\ast$. We find infinitely many $3$-dimensional counterexamples, as well as another infinite family of flat counterexamples whose dimensions grow without bound.
A biquotient vector bundle is any vector bundle over a biquotient $G/\!\!/ H$ of the form $G\times_{H} V$ for an $H$-representation $V$. Over most biquotients, biquotient vector bundles are the only vector bundles known to admit metrics of non-negative sectional curvature, and hence they play a crucial role in the context of the converse to the Soul Theorem of Cheeger and Gromoll. In this article, we study the question of when the tangent bundle of $G/\!\!/ H$ is a biquotient vector bundle. We find infinite families of examples of biquotients $M\cong G/\!\!/ H$ for which the tangent bundle is not a biquotient vector bundle for any presentation of $M$ as a biquotient. In addition, we find infinite families of manifolds which arise as biquotients in two ways: one for which the tangent bundle is a biquotient bundle, and one for which it is not. Some of these results depend on an observation of Hirzebruch which relates the signature and Euler characteristic of an almost complex manifold. We include a proof of this fact as it seems to be missing from the literature.
We classify cohomogeneity one actions on smooth, simply connected, closed manifolds with the rational cohomology of a sphere. In particular, we show that such a manifold is diffeomorphic to a sphere, a Brieskorn variety, the Wu manifold $SU(3)/SO(3)$, one of two homogeneous spaces of the form $\mathbf{G}_2/SU(2)$,or a member of a particular four-parameter family of $7$-manifolds.
We completely characterize the sectional curvature of all of the $13$-dimensional Bazaikin spaces. In particular, we show that all Bazaikin spaces admit a quasi-positively curved Riemannian metric, and that, up to isometry, there is a unique Bazaikin space which is almost positively curved but not positively curved.
In previous work, the second author and others have found conditions on a homogeneous space $G/H$ which imply that, up to stabilization, all vector bundles over $G/H$ admit Riemannian metrics of non-negative sectional curvature. One important ingredient of their approach is Segal's result that the set of vector bundles of the form $G\times_H V$ for a representation $V$ of $H$ contains inverses within the class. We show that this approach cannot work for biquotients $G/\!\!/ H$, where we consider vector bundles of the form $G\times_{H} V$. We call such vector bundles biquotient bundles. Specifically, we show that in each dimension $n\geq 4$ except $n=5$, there is a simply connected biquotient of dimension $n$ with a biquotient bundle which does not contain an inverse within the class of biquotient bundles. In addition, we show that for $n\geq 6$ except $n=7$, there are infinitely many homotopy types of biquotients with the property that no non-trivial biquotient bundle has an inverse. Lastly, we show that every biquotient bundle over every simply connected biquotient $M^n = G/\!\!/ H$ with $G$ simply connected and with $n\in \{2,3,5\}$ has an inverse in the class of biquotient bundles.
A manifold $M$ is said to be a double disk bundle if it can be decomposed as a union of two disk bundles glued together by a diffeomorphism of their boundaries. We show that if $M^n$ is a closed simply connected $n$-manifold with $n$ even which is simultaneously a double disk bundle and a rational homology sphere, then $M$ must be homeomorphic to a sphere. In addition, we show that in any dimension, if $M$ is a highly connected rational homology sphere which supports a double disk bundle structure, then its "middle" cohomlogy group must be cyclic.
For $G$ a simple, connected graph, a vertex labeling $f:V(G)\rightarrow \mathbb{Z}_+$ is called a $\textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\geq \operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,v\in V(G)$. The $\textit{radio number}$ of $G$ is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,...,|V(G)|\}$ exists, $G$ is called a $\textit{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.
Under mild topological restrictions, this article establishes that a smooth, closed, simply connected manifold of dimension at most seven which can be decomposed as the union of two disk bundles must be rationally elliptic. In dimension five, such manifolds are classified up to diffeomorphism, while the same is true in dimension six when either the second Betti number vanishes or the third Betti number is non-trivial.
Each series $\sum_{n=1}^\infty a_n$ of real positive terms gives rise to a topology on $\mathbb{N} = \{1,2,3,...\}$ by declaring a proper subset $A\subseteq \mathbb{N}$ to be closed if $\sum_{n\in A} a_n < \infty$. We explore the relationship between analytic properties of the series and topological properties on $\mathbb{N}$. In particular, we show that, up to homeomorphism, $|\mathbb{R}|$-many topologies are generated. We also find an uncountable family of examples $\{\mathbb{N}_\alpha\}_{\alpha \in [0,1]}$ with the property that for any $\alpha < \beta$, there is a continuous bijection $\mathbb{N}_\beta\rightarrow \mathbb{N}_\alpha$, but the only continuous functions $\mathbb{N}_\alpha\rightarrow \mathbb{N}_\beta$ are constant.
We classify simply connected, closed cohomogeneity one manifolds with singly generated or 4-periodic rational cohomology and positive Euler characteristic.
An $n$-dimensional manifold $M$ is said to be rationally $4$-periodic if there is an element $e\in H^4(M;\mathbb{Q})$ with the property that cupping with $e$, $\cdot \cup e:H^\ast(M;\mathbb{Q})\rightarrow H^{\ast + 4}(M;\mathbb{Q})$ is injective for $0< \ast \leq \dim M-4$ and surjective when $0\leq \ast < \dim M-4$. We classify all compact simply connected biquotients which are rationally $4$-periodic. In addition, we show that if a simply connected rationally elliptic CW-complex $X$ of dimension at least $6$ is rationally $4$-periodic, then the cohomology ring is either singly generated, or $X$ is rationally homotopy equivalent to $S^2\times \mathbb{H}P^n$, $S^3\times \mathbb{H}P^n$, or $S^3\times S^3$.
A Riemannian manifold is said to be almost positively curved if the sets of points for which all $2$-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than $1$. The construction and verification rely on the Lie group $\mathbf{G}_2$ and the octonions, so do not obviously generalize to any other Grassmannians.