Mathematics

In this paper, we attempt to use two types of flows to study the relations between quermassintegrals A k (see Definition 1.1), which correspond to the Alexandrov-Fenchel inequalities for closed convex C 2 -hypersurfaces in S n+1 + .

In this paper we characterize the Einstein metrics in such broader classes of metrics as almost η -Ricci solitons and η -Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an η -Ricci soliton is Einstein metric if either it is η -Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost η -Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of η -Ricci solitons and gradient η -Ricci solitons, which illustrate our results.

We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds.

We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy--Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.

The moduli space of Higgs bundles can be defined as a quotient of an infinite-dimensional space. Moreover, by the Kuranishi slice method, it is equipped with the structure of a normal complex space. In this paper, we will use analytic methods to show that the moduli space is quasi-projective. In fact, following Hausel's method, we will use the symplectic cut to construct a normal and projective compactification of the moduli space, and hence prove the quasi-projectivity. The main difference between this paper and Hausel's is that the smoothness of the moduli space is not assumed.

The stable reduction theorem says that a family of curves of genus g≥2 over a punctured curve can be uniquely completed (after possible base change) by inserting certain stable curves at the punctures. We give a new proof of this result for curves defined over C using the Kähler-Einstein metrics on the fibers to obtain the limiting stable curves at the punctures.

The critical CR GJMS operator on a strictly pseudoconvex CR manifold is a non-hypoelliptic CR invariant differential operator. We prove that, under the embeddability assumption, it is essentially self-adjoint and has closed range. Moreover, its spectrum is discrete, and the eigenspace corresponding to each non-zero eigenvalue is a finite-dimensional subspace of the space of smooth functions. As an application, we obtain a necessary and sufficient condition for the existence of a contact form with zero CR Q -curvature.

We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type 1 4 (1,1) . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type %of dimension 10−k associated with a unimodular lattice of signature (2,10−k) . We prove that the invariant, viewed as a function on the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension 2 . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.

We show an equality between the analytic torsion and the absolute value at the zero point of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author's previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.

For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on M , and by a result of Böhm-Lafuente-Simon ([BöLS17]) they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b 2 (M)=1 , for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α -limit set consists of fixed points at infinity of M G . Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system's phase space.