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Register a new userOn compactness conformally compact Einstein manifolds and uniqueness of Graham-Lee metrics, III
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In this paper, we establish a compactness result for a class of conformally compact Einstein metrics defined on manifolds of dimension $dge 4$. As an application, we derive the global uniqueness of a class of conformally compact Einstein metric defined on the $d$-dimensional ball constructed in the earlier work of Graham-Lee with $dge 4$. As a second application, we establish some gap phenomenon for a class of conformal invariants.
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We classify quadruples $(M,g,m,tau)$ in which $(M,g)$ is a compact Kahler manifold of complex dimension $m>2$ with a nonconstant function $tau$ on $M$ such that the conformally related metric $g/tau^2$, defined wherever $tau e 0$, is Einstein. It turns out that $M$ then is the total space of a holomorphic $CP^1$ bundle over a compact Kahler-Einstein manifold $(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with Kahler metrics $g$ that are locally reducible; a second, discovered by Berard Bergery (1982), and having $tau e 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kahler surface metrics; and a fourth family, present only in odd complex dimensions $mge 9$. Our classification uses a {it moduli curve}, which is a subset $mathcal{C}$, depending on $m$, of an algebraic curve in $R^2$. A point $(u,v)$ in $mathcal{C}$ is naturally associated with any $(M,g,m,tau)$ having all of the above properties except for compactness of $M$, replaced by a weaker requirement of ``vertical compactness. One may in turn reconstruct $M,g$ and $tau$ from this $(u,v)$ coupled with some other data, among them a Kahler-Einstein base $(N,h)$ for the $CP^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m,tau)$ with compact $M$ form a countably infinite subset of $mathcal{C}$.
In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.
Let $mathcal{K}(n, V)$ be the set of $n$-dimensional compact Kahler-Einstein manifolds $(X, g)$ satisfying $Ric(g)= - g$ with volume bounded above by $V$. We prove that after passing to a subsequence, any sequence ${ (X_j, g_j)}_{j=1}^infty$ in $mathcal{K}(n, V)$ converges, in the pointed Gromov-Hausdorff topology, to a finite union of complete Kahler-Einstein metric spaces without loss of volume. The convergence is smooth off a closed singular set of Hausdorff dimension no greater than $2n-4$, and the limiting metric space is biholomorphic to an $n$-dimensional semi-log canonical model with its non log terminal locus of complex dimension no greater than $n-1$ removed. We also show that the Weil-Petersson metric extends uniquely to a Kahler current with bounded local potentials on the KSBA compactification of the moduli space of canonically polarized manifolds. In particular, the coarse KSBA moduli space has finite volume with respect to the Weil-Petersson metric. Our results are a high dimensional generalization of the well known compactness results for hyperbolic metrics on compact Riemann surfaces of fixed genus greater than one.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${rm Ric} = 2 F^2$, ${rm Ric}=0$ and ${rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
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