My research is in Riemannian geometry. You can find me on the arXiv, Google Scholar, or MathSciNet (Author ID is 852611).
My work has been supported by the National Science Foundation grants DMS-1404670/1622541 (2014 - 2017) and DMS-1708493 (2017 - 2020).
Preprints:
- L. Kennard, W. Wylie, D. Yeroshkin. The weighted connection and sectional curvature for manifolds with density (preprint).
Abstract: In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.
- M. Amann, L. Kennard. Positive curvature and symmetry in small dimensions, I (preprint).
Abstract: This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds in dimensions up to 16. Many of the calculations are sharp and they require less symmetry than previous classifications. When restricted to certain classes of manifolds that admit non-negative curvature, these results imply diffeomorphism classifications. Also studied is a closely related family of manifolds called positively elliptic manifolds, and we prove the Halperin conjecture in this context for dimensions up to 16 or Euler characteristics up to 16.
Publications:
- L. Kennard, Z. Su. On dimensions supporting a rational projective plane.
J. Topol. Anal., to appear (preprint).
Abstract: A rational projective plane (QP^2) is a simply connected, smooth, closed manifold M such that H^*(M;Q) = Q[x]/(x^3). An open problem is to classify the dimensions at which such a manifold exists. The Barge-Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori-Stong integrality conditions on the Pontryagin numbers. In this article, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a QP^2. We then confirm existence of a QP^2 in two new dimensions and prove several non-existence results using factorizations of numerators of divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces.
- L. Kennard. Fundamental groups of manifolds with positive sectional curvature and torus symmetry.
J. Geom. Anal., vol. 27 (2017), pp. 2894-2925.
Abstract: In 1965, S.-S. Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories.
- M. Amann, L. Kennard. On a generalized conjecture of Hopf with symmetry.
Compos. Math., vol. 153 (2017), pp. 313-322.
Abstract: A famous conjecture of Hopf is that the product of the two-dimensional sphere with itself does not admit a Riemannian metric with positive sectional curvature. More generally, one may conjecture that this holds for any nontrivial product. We provide evidence for this generalized conjecture in the presence of symmetry.
- L. Kennard, W. Wylie. Positive weighted sectional curvature.
Indiana Univ. Math. J., vol. 66 (2017), no. 2, pp. 419-462.
Abstract: In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's formula for submersions, Frankel's theorem, and Wilking's connectedness lemma. As applications of these results, we recover weighted versions of topological classification results of Grove-Searle and Wilking for manifolds of high symmetry rank and positive curvature.
- M. Amann, L. Kennard. Positive curvature and rational ellipticity.
Algebr. Geom. Topol., vol. 15 (2015), no. 4, pp. 2269-2301.
Abstract: Simply-connected manifolds of positive sectional curvature M are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.
- M. Amann, L. Kennard. Topological properties of positively curved manifolds with symmetry.
Geom. Funct. Anal., vol. 24 (2014), no. 5, pp. 1377-1405.
Abstract: We obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.
- L. Kennard. On the Hopf conjecture with symmetry.
Geometry of manifolds with non-negative sectional curvature, Lecture Notes in Math., vol. 2110 (2014), pp. 111-116.
Abstract: The Hopf conjecture states that an even-dimensional manifold with positive curvature has positive Euler characteristic. We show that this is true under the assumption that a torus of sufficiently large dimension acts by isometries. This improves previous results by replacing linear bounds by a logarithmic bound. The new method that is introduced is the use of Steenrod squares combined with geometric arguments of a similar type to what was done before.
- L. Kennard. Positively curved Riemannian manifolds with logarithmic symmetry rank bounds.
Comm. Math. Helv., vol. 89 (2014), no. 4, pp. 937-962.
Abstract: We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group.
- L. Kennard. On the Hopf conjecture with symmetry.
Geom. Topol. vol. 161 (2013), no. 1, pp. 563-593.
Abstract: The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new tool is the action of the Steenrod algebra on cohomology.
Publications involving undergraduates (asterisk denotes student):
- L. Kennard, J. Rainone*. Characterizations of the round two-dimensional sphere in terms of closed geodesics.
Involve, to appear (preprint).
Abstract: The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.
- J. Holdener, L. Kennard*, M. Zaremsky*. Generalized Thue-Morse sequences and the von Koch curve. Int. J. Pure Appl. Math. vol. 47 (2008), no. 3, pp. 397-403. MathSciNet: MR2458634.
Abstract: In a recent paper, Ma and Holdener used turtle geometry and polygon maps to show that the Thue-Morse sequence encodes the von Koch curve. In the final paragraph of this same paper, they ask whether or not there exist certain generalized Thue-Morse sequences that also encode the curve. Here we answer this question in the affirmative, providing an infinite family of words that generate generalized Thue-Morse sequences encoding the von Koch curve.
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