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Open Access Publications from the University of California

Scholarly Works (12 results)

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UCLA Previously Published Works (2022)
AbstractWe prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence-free Weyl tensors and "Equation missing"-nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence-free Bochner tensors. For quaternion Kähler manifolds, we obtain a partial result towards the LeBrun–Salamon conjecture.
    Cover page: Tachibana-type theorems and special Holonomy
    • Thesis
    • Peer Reviewed
    UCLA Electronic Theses and Dissertations (2012)

    This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is n -1, then the metric is a warped product where the base is an open interval. Tojeiro generalized this result given the existence of a certain type of function of the eigenvalues of the Codazzi tensor, of which the trace is one example. We will show the equivalence of Tojeiro's condition to three other conditions. Furthermore, we construct examples of Codazzi tensors

    having two eigenvalue functions, one of which has eigenspace dimension n-1 where the metric is not a warped product with interval base, refuting a remark in Besse that the warped product conclusion holds without any restriction on the trace.

      Cover page: Codazzi Tensors with Two Eigenvalue Functions
      • Article
      • Peer Reviewed
      UCLA Previously Published Works (2021)
      Abstract We show that compact Kähler manifolds have the rational cohomology ring of complex projective space provided a weighted sum of the lowest three eigenvalues of the Kähler curvature operator is positive. This follows from a more general vanishing and estimation theorem for the individual Hodge numbers. We also prove an analogue of Tachibana’s theorem for Kähler manifolds.
        Cover page: Vanishing and estimation results for Hodge numbers
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        UCLA Previously Published Works (2022)
          Cover page: Alexandrov spaces with maximal radius
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          UCLA Previously Published Works (2020)
            Cover page: The Bochner Technique and Weighted Curvatures
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            UCLA Previously Published Works (2017)
              Cover page: A step towards the Alekseevskii conjecture
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              UCLA Previously Published Works (2023)
              Abstract: We show that compact, ‐dimensional Riemannian manifolds with ‐nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the th Betti number provided that the curvature operator of the second kind is ‐positive. Our curvature conditions become weaker as increases. For , we have , and for , we exhibit a ‐positive algebraic curvature operator of the second kind with negative Ricci curvatures.
                Cover page: Betti numbers and the curvature operator of the second kindCreative Commons 'BY' version 4.0 license
                • Article
                • Peer Reviewed
                UCLA Previously Published Works (2014)
                In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.
                  Cover page: Warped product Einstein metrics over spaces with constant scalar curvature
                  • Article
                  • Peer Reviewed
                  UCLA Previously Published Works (2015)
                  Abstract In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that every normal semi-algebraic Ricci soliton also admits a k-dimensional Einstein extension for any k ≥ 2. We also prove converse theorems for these constructions and some geometric and topological structure results for homogeneous warped product Einstein metrics. In the appendix we give an alternative approach to semi-algebraic Ricci solitons which naturally leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous setting.
                    Cover page: Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons
                    • Article
                    • Peer Reviewed
                    UCLA Previously Published Works (2016)
                    Abstract We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all solvable Einstein metrics of dimension seven or less with codimension-one nilradical. For each linearly stable metric, dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.
                      Cover page: Linear stability of algebraic Ricci solitons
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