Almost Hermitian Ricci flow
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
math.DG↗
arXiv subjects
Casey Lynn Kelleher
Publications and source records attributed to Casey Lynn Kelleher.
Index-energy estimates for Yang-Mills connections and Einstein metrics
We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature.
math.DG↗
Symplectic curvature flow revisited
We continue studying a parabolic flow of almost Kähler structures introduced by Streets and Tian which naturally extends Kähler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex structure, Chern torsion and Chern connection, we establish new formulas for the evolutions of canonical quantities, in particular those related to the Chern connection. Using this, we give an extended characterization of fixed points of the flow originally performed by Streets and Tian.
math.DG↗
A conformally invariant gap theorem in Yang-Mills theory
We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.
math.DG↗
Limits of Yang-Mills α-connections
In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small α values the SO(4) invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold.
math.DG↗
Singularity formation of the Yang-Mills flow
We study singularity structure of Yang-Mills flow in dimensions $n \geq 4$. First we obtain a description of the singular set in terms of concentration for a localized entropy quantity, which leads to an estimate of its Hausdorff dimension. We develop a theory of tangent measures for the flow, which leads to a stratification of the singular set. By a refined blowup analysis we obtain Yang-Mills connections or solitons as blowup limits at any point in the singular set.
math.DG↗
Entropy, stability, and harmonic map flow
Inspired by work of Colding-Minicozzi on mean curvature flow, Zhang introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable $\mathcal F$-stability. Then, focusing on the case of spherical targets, we prove a general instability result for high-entropy solitons. Finally, we exploit results of Lin-Wang to observe long time existence and convergence results for maps into certain convex domains and how they relate to generic singularities of harmonic map flow.
math.DG↗
Higher order Yang-Mills flow
We define a family of functionals generalizing the Yang-Mills functional. We study the corresponding gradient flows and prove long-time existence and convergence results for subcritical dimensions as well as a bubbling criterion for the critical dimensions. Consequently, we have an alternate proof of the convergence of Yang-Mills flow in dimensions 2 and 3 given by Rade and the bubbling criterion in dimension 4 of Struwe in the case where the initial flow data is smooth.
math.DG↗
Asymptotic expansion of the Bergman kernel via perturbation of the Bargmann-Fock model
We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential $kφ$ in a $\frac{1}{\sqrt{k}}$-neighborhood of a given point, the potential becomes an asymptotic perturbation of the Bargmann-Fock metric. We then prove that the Bergman kernel is also an asymptotic perturbation of the Bargmann-Fock Bergman kernel.
math.DG↗
Entropy, Stability, and Yang-Mills flow
Following work of Colding-Minicozzi, we define a notion of entropy for connections over $\mathbb R^n$ which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stability in terms of the spectrum of a certain linear operator associated to the soliton. This leads furthermore to a gap theorem for solitons. These results point to a broader strategy of studying "generic singularities" of Yang-Mills flow, and we discuss the differences in this strategy in dimension $n=4$ versus $n \geq 5$.
math.DG↗
Polygonal equalities and virtual degeneracy in $L_{p}$-spaces
Suppose $0 < p \leq 2$ and that $(Ω, μ)$ is a measure space for which $L_{p}(Ω, μ)$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_{p}(Ω, μ)$ that have strict $p$-negative type. In order to do this we study non-trivial $p$-polygonal equalities in $L_{p}(Ω, μ)$. These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form \begin{eqnarray*} \sum\limits_{j, i = 1}^{n} α_{j} α_{i} {\| z_{j} - z_{i} \|}_{p}^{p} & = & 0 \end{eqnarray*} where $\{ z_{1}, \ldots, z_{n} \}$ is a subset of $L_{p}(Ω, μ)$ and $α_{1}, \ldots, α_{n}$ are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial $p$-polygonal equalities in $L_{p}(Ω, μ)$. The cases $p < 2$ and $p = 2$ are substantially different and are treated separately. The case $p = 1$ generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial $p$-polygonal equalities in $L_{p}(Ω, μ)$ is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if $(X,d)$ is a metric space that has strict $q$-negative type for some $q \geq p$, then: (1) $(X,d)$ is not isometric to any linear subspace $W$ of $L_{p}(Ω, μ)$ that contains a pair of disjointly supported non-zero vectors, and (2) $(X,d)$ is not isometric to any subset of $L_{p}(Ω, μ)$ that has non-empty interior. Furthermore, in the case $p = 2$, it also follows that $(X,d)$ is not isometric to any affinely dependent subset of $L_{2}(Ω, μ)$.
math.FA↗
Strongly non embeddable metric spaces
Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(\mathfrak{Z}, ζ)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $\mathbb{Z}_ω=\bigoplus_{\aleph_0}\mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p \geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z \to \ell_{p}$.
math.FA↗