1.1. Log-K-stability Let (X, J) be a Fano manifold, that is, K−1 X is ample. A basic problem in Kähler geometry is to determine whether (X, J) has a Kähler–Einstein metric (see [22]). The existence problem of Kähler–Einstein metric is a special case of the existence problem of constant scalar curvature Kähler (cscK) metric. For the latter, we fix an ample line bundle L on (X, J). We have the fo...

We consider a timelike geodesic congruence in the presence of perturbative quantum fluctuations spacetime metric. calculate change volume bundle geodesics due to such and thereby obtain quantum-gravitationally modified Raychaudhuri equation. Quantum gravity generically increases convergence congruences production caustics.

A new 8-dimensional conformal gauging avoids the unphysical size change, third order gravitational field equations, and auxiliary fields that prevent taking the conformal group as a fundamental symmetry. We give the structure equations, gauge transformations and intrinsic metric structure for the new biconformal spaces. We prove that a torsion-free biconformal space with exact Weyl form, closed...

Let F be a flat vector bundle over a compact Riemannian manifold M and let f : M → R be a Morse function. Let g be a smooth Euclidean metric on F , let g t = e g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g t . Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ρ(t) for t → +∞ of the form a0 + a1t +...

Let $N$ be a closed manifold and $U \subset T^\*(N)$ bounded domain in the cotangent bundle of $N$, containing zero-section. A conjecture due to Viterbo asserts that spectral metric for Lagrangian submanifolds $U$ are exact-isotopic zero-section is bounded. In this paper we establish an upper bound on distance between two such Lagrangians $L\_0, L\_1$, which depends linearly boundary depth Floe...

In their recent paper [8], Kulharni and Raymond show that a closed 3-manifold which admits a complete Lorentz metric of constant curvature 1 (henceforth called a complete Lorentz structure) must be Seifert fibered over a hyperbolic base. Furthermore on every such Seifert fibered 3-manifold with nonzero Euler class they construct such a Lorentz metric. Moreover the Lorentz structure they constru...

The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits hermitian metric with positive curvature in the sense of Griffiths. In this article we give sufficient condition for on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce $L^2$-metric $E$. This result suggests study relative K\ahler-Ricci flow fibration $\mathbb{P}(E^*)\to S$. We define and ...

Using Monge-Amp\`ere geometry, we study the singular structure of a class nonlinear equations in three dimensions, arising geophysical fluid dynamics. We extend seminal earlier work on geometry by examining role an induced metric Lagrangian submanifolds cotangent bundle. In particular, show that signature serves as classification equation, while singularities and elliptic-hyperbolic transitions...