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Register a new userOn Variational Properties of Quadratic Curvature Functionals
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In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold $M$ of dimension $nge 3$ on the space of Riemannian metrics on $M$ with unit volume. We study the stability of these functionals at the metric with constant sectional curvature as its critical point.
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In this article we continue the study of the two curvature notions for Kahler manifolds introduced by the first named author earlier: the so-called cross quadratic bisectional curvature (CQB) and its dual ($^d$CQB). We first show that compact Kahler manifolds with CQB$_1>0$ or $mbox{}^d$CQB$_1>0$ are Fano, while nonnegative CQB$_1$ or $mbox{}^d$CQB$_1$ leads to a Fano manifold as well, provided that the universal cover does not contain a flat de Rham factor. For the latter statement we employ the Kahler-Ricci flow to deform the metric. We conjecture that all Kahler C-spaces will have nonnegative CQB and positive $^d$CQB. By giving irreducible such examples with arbitrarily large second Betti numbers we show that the positivity of these two curvature put no restriction on the Betti number. A strengthened conjecture is that any Kahler C-space will actually have positive CQB unless it is a ${mathbb P}^1$ bundle. Finally we give an example of non-symmetric, irreducible Kahler C-space with $b_2>1$ and positive CQB, as well as compact non-locally symmetric Kahler manifolds with CQB$<0$ and $^d$CQB$<0$.
We provide a general Bochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.
Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{round})$. If $p>frac{m}{2}$ and $int_{M} left{ Rc-(m-1)gright}_{-}^{p} dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^{m}, g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.
We examine relations between geometry and the associated curvature decompositions in Weyl geometry.
Following the global method for relaxation we prove an integral representation result for a large class of variational functionals naturally defined on the space of functions with Bounded Deformation. Mild additional continuity assumptions are required on the functionals.
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