Browsing by Author "Gao, Zhiyong"

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    A minimization of a curvature functional on fiber bundles
    (1998) Hawkins, Christopher Ryan; Gao, Zhiyong
    Let B be a smooth compact orientable surface without boundary and with $\chi(B) < 0.$ We examine two types of fiber bundles M over B with fiber F. The first is a principle fiber bundle with a two-torus fiber and the second is an $S\sp2$ fiber bundle with an SO(3) group action. In each case, the tangent space of the bundle can be decomposed into a vertical space, those vectors tangent to fibers, and a horizontal space complementary to the vertical space and invariant under the group action. The bundle can be given a metric that is the direct sum of metrics on the vertical and horizontal spaces. Additionally, with this metric, M, is locally isometric to a product space $B\times F$ with metric $g\sb{b} + g\sb{f}.$ Here $g\sb{b}$ is any fixed metric on the base, $g\sb{f}$ is a constant curvature metric on the fiber invariant under the action of the group. We can obtain a new metric on M by scaling the horizontal component of the original by $e\sp{2u}$ and the vertical component by $f\sp2,$ where u and f are smooth functions on the base. We put certain constraints on u and f and consider the family of all such variations. In this thesis, we show, using nonlinear elliptic estimates, that among these metrics there is one for which the integral of the norm of the Ricci curvature tensor squared, $\int\sb{M}\vert Ric\vert\sp2dV,$ is minimized.
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    Critical Riemannian metrics
    (1990) Chang, Shun-Cheng; Gao, Zhiyong
    Let $(M,g)$ be a compact oriented n-dimensional smooth Riemannian manifold. Consider the following quadratic Riemannian functional$$SR(g) = \int\sb{M}\ \vert R\sb{ijkl}(g)\vert \sp{2}d\mu$$which is homogeneous of degree ${n\over2}-2,$ where $R\sb{ijkl}$(g) is the curvature tensors of $(M,g)$ and $d\mu$ is the volume element measured by g. A critical point of $SR(g)$ is called a critical metric on M, that is, the Ricci tensor satisfies the critical equations grad$SR\sb{g}$ = 0. In particular, for a compact 4-manifold M, every Einstein metric is a critical metric for SR on M. In this thesis, we propose an extension of the compactness property for Einstein metrics to critical metrics on a compact smooth Riemannian 4-manifold M. More precisely, first we consider the subspace $G(M)$ of all critical metrics on M with the injectivity radius bounded from below by a constant $i\sb{0} >$ 0 and diameter bounded from above by d. Then we are able to prove that $G(M)$ is compact as a subset of moduli space of critical metrics in the $C\sp{\infty}$-topology (Theorem 6.1). Second, we replaced the injectivity radius lower bound by the local volume bound, then we get a compact 4-dimensional critical orbifold (Theorem 7.1). Furthermore, by using the fundamental equations of Riemannian submersions with totally geodesic fibers, we construct some critical Riemannian 4-manifolds.
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    Harmonic diffeomorphisms between manifolds with bounded curvature
    (1991) Anderson, John Patrick; Gao, Zhiyong
    Let compact n-dimensional Riemannian manifolds $(M,g),\ (\widehat M,\ g)$ a diffeomorphism $u\sb0: M\to \widehat M,$ and a constant $p > n$ be given. Then sufficiently small $L\sp{p}$ bounds on the curvature of $\widehat M$ and on the difference of $g$ and $u\sbsp{0}{\*}\ g$ guarantee that $u\sb0$ can be continuously deformed to a harmonic diffeomorphism. A vector field $v$ is constructed on the space of mappings $u$ which are $L\sp{2,p}$ close to $u\sb0$ by solving the nonlinear elliptic equation $\Delta v + \widehat{Rc}\ v = -\Delta u.$ It is shown that under sufficient conditions on $u\sb0$ and on the curvature $\widehat{Rm}$ of the target, the integral curve $u\sb t$ of this vector field converges to a harmonic diffeomorphism. Since the objects we work with, such as $v$ and its derivatives, live in bundles over $M$, to prove regularity results we must first adapt standard techniques and results of elliptic theory to the bundle case. Among the generalizations we prove are Moser iteration, a Sobolev embedding theorem, and a Calderon-Zygmund inequality.